Package 'BMT'

The BMT Distribution.

Description

Density, distribution, quantile function, random number generation for the BMT distribution, with p3 and p4 tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

Usage

dBMT(x, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d",
  log = FALSE)

pBMT(q, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d",
  lower.tail = TRUE, log.p = FALSE)

qBMT(p, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d",
  lower.tail = TRUE, log.p = FALSE)

rBMT(n, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

Arguments

x, q	vector of quantiles.
p3, p4	tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT distribution.
type.p.3.4	type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
p1, p2	domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution.
type.p.1.2	type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the lenght is taken to be the number required

Details

The BMT distribution with tails weights and domain parametrization (type.p.3.4 = "t w" and type.p.1.2 = "c-d") has quantile function

$(d - c) [3 t_p ( 1 - t_p )^2 \kappa_l - 3 t_p^2 ( 1 - t_p ) \kappa_r + t_p^2 ( 3 - 2 t_p ) ] + c$

where $0 \le p \le 1$, $t_p = 1/2 - \cos ( [\arccos ( 2 p - 1 ) - 2 \pi] / 3 )$, and $0 < \kappa_l < 1$ and $0 < \kappa_r < 1$ are, respectively, related to left and right tail weights or curvatures.

The BMT coefficient of asymmetry $-1 < \zeta < 1$ is

$\kappa_r - \kappa_l$

The BMT coefficient of steepness $0 < \xi < 1$ is

$(\kappa_r + \kappa_l - |\kappa_r - \kappa_l|) / (2 (1 - |\kappa_r - \kappa_l|))$

for $|\kappa_r - \kappa_l| < 1$.

Value

dBMT gives the density, pBMT the distribution function, qBMT the quantile function, and rBMT generates random deviates.

The length of the result is determined by n for rBMT, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

If type.p.3.4 == "t w", p3 < 0 and p3 > 1 are errors and return NaN.

If type.p.3.4 == "a-s", p3 < -1 and p3 > 1 are errors and return NaN.

p4 < 0 and p4 > 1 are errors and return NaN.

If type.p.1.2 == "c-d", p1 >= p2 is an error and returns NaN.

If type.p.1.2 == "l-s", p2 <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected] and Alvaro Mauricio Montenegro Diaz [ths]

References

Torres-Jimenez, C. J. and Montenegro-Diaz, A. M. (2017, September), An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

BMTcentral, BMTdispersion, BMTskewness, BMTkurtosis, BMTmoments for descriptive measures or moments. BMTchangepars for parameter conversion between different parametrizations.

Examples

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
z <- seq(0, 1, length.out = 100)
F1 <- pBMT(z, 0.25, 0.75, "t w")
Q1 <- qBMT(F1, 0.25, 0.75, "t w")
max(abs(z - Q1))
f1 <- dBMT(z, 0.25, 0.75, "t w")
r1 <- rBMT(100, 0.25, 0.75, "t w")
layout(matrix(c(1,2,1,3), 2, 2))
hist(r1, freq = FALSE, xlim = c(0,1))
lines(z, f1)
plot(z, F1, type="l")
plot(F1, Q1, type="l")

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.5
F2 <- pBMT(z, 0.5, 0.5, "a-s")
Q2 <- qBMT(F2, 0.5, 0.5, "a-s")
f2 <- dBMT(z, 0.5, 0.5, "a-s")
r2 <- rBMT(100, 0.5, 0.5, "a-s")
max(abs(f1 - f2))
max(abs(F1 - F2))
max(abs(Q1 - Q2))

# BMT on [-1.783489, 3.312195] with 
# left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
x <- seq(-1.783489, 3.312195, length.out = 100)
F3 <- pBMT(x, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
Q3 <- qBMT(F3, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
max(abs(x - Q3))
f3 <- dBMT(x, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
r3 <- rBMT(100, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
layout(matrix(c(1,2,1,3), 2, 2))
hist(r3, freq = FALSE, xlim = c(-1.783489,3.312195))
lines(x, f3)
plot(x, F3, type="l")
plot(F3, Q3, type="l")

# BMT with mean equal to 0, standard deviation equal to 1, 
# asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.5
f4 <- dBMT(x, 0.5, 0.5, "a-s", 0, 1, "l-s")
F4 <- pBMT(x, 0.5, 0.5, "a-s", 0, 1, "l-s")
Q4 <- qBMT(F4, 0.5, 0.5, "a-s", 0, 1, "l-s")
r4 <- rBMT(100, 0.5, 0.5, "a-s", 0, 1, "l-s")
max(abs(f3 - f4))
max(abs(F3 - F4))
max(abs(Q3 - Q4))

---

The BMT-Phi Distribution.

Description

Density, distribution function, quantile function, random number generation for the BMT-Phi distribution with mean equal to mean and standard deviation equal to sd.

Usage

dBMT.Phi(x, mean = 0, sd = 1, log = FALSE)

pBMT.Phi(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

qBMT.Phi(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

rBMT.Phi(n, mean = 0, sd = 1)

Arguments

x, q	vector of quantiles.
mean	vector of means.
sd	vector of standard deviations.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the lenght is taken to be the number required

Details

If mean or sd are not specified they assume the default values of 0 and 1, respectively.

The BMT-Phi distribution is the BMT distribution with $\kappa_l = \kappa_r = 0.58029164978583758$. The BMT-Phi cumulative distribution function (cdf) is the closest BMT cdf to the normal cdf with the same mean and standard deviation.

Value

dBMT.Phi gives the density, pBMT.Phi the distribution function, qBMT.Phi the quantile function, and rBMT.Phi generates random deviates.

The length of the result is determined by n for rBMT.Phi, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

sd <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

References

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

Distributions for other standard distributions. pBMT for the BMT distribution and pBMT.Psi for the BMT-Psi distribution.

Examples

layout(matrix(1:4,2,2))

curve(pnorm(x), -4, 4, col = "red", lty = 2, ylab = "cdf")
curve(pBMT.Phi(x), add = TRUE, col = "blue", lty = 3)
legend("topleft", legend = c("norm(0,1)","BMT-Phi(0,1)"), 
       bty = "n", col = c("red","blue"), lty = 2:3)

curve(pnorm(x)-pBMT.Phi(x), -4, 4)

curve(qnorm(x), col = "red", lty = 2, xlab = "p", ylab = "qf")
curve(qBMT.Phi(x), add = TRUE, col = "blue", lty = 3)

hist(rBMT.Phi(10000), freq = FALSE, breaks = seq(-4,4,0.25), border = "blue")
curve(dnorm(x), add = TRUE, col = "red", lty = 2)
curve(dBMT.Phi(x), add = TRUE, col = "blue", lty = 3)

---

The BMT-Psi Distribution.

Description

Density, distribution function, quantile function, random number generation for the BMT-Psi distribution with mean equal to mean and standard deviation equal to sd.

Usage

dBMT.Psi(x, mean = 0, sd = 1, log = FALSE)

pBMT.Psi(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

qBMT.Psi(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

rBMT.Psi(n, mean = 0, sd = 1)

Arguments

x, q	vector of quantiles.
mean	vector of means.
sd	vector of standard deviations.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the lenght is taken to be the number required

Details

If mean or sd are not specified they assume the default values of 0 and 1, respectively.

The BMT-Psi distribution is the BMT distribution with $\kappa_l = \kappa_r = 0.63355781127887611515$. The BMT-Psi cumulative distribution function (cdf) is the closest BMT cdf to the logistic cdf with scale = 1 / d and d = 1.70174439 (Camilli, 1994, p. 295).

Value

dBMT.Psi gives the density, pBMT.Psi the distribution function, qBMT.Psi the quantile function, and rBMT.Psi generates random deviates.

The length of the result is determined by n for rBMT.Psi, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

sd <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

References

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

Camilli, G. (1994). Teacher's corner: origin of the scaling constant d= 1.7 in item response theory. Journal of Educational Statistics, 19(3), 293-295.

See Also

Distributions for other standard distributions. pBMT for the BMT distribution and pBMT.Phi for the BMT-Phi distribution.

Examples

layout(matrix(1:4, 2, 2))

curve(plogis(x, scale = 1 / 1.70174439), -4, 4, col = "red", lty = 2, ylab = "cdf")
curve(pBMT.Psi(x), add = TRUE, col = "blue", lty = 3)
legend("topleft", legend = c("logis(0, 1 / 1.70174439)","BMT-Psi(0,1)"), 
       bty = "n", col = c("red","blue"), lty = 2:3)

curve(plogis(x, scale = 1 / 1.70174439)-pBMT.Psi(x), -4, 4)

curve(qlogis(x, scale = 1 / 1.70174439), col = "red", lty = 2, xlab = "p", ylab = "qf")
curve(qBMT.Psi(x), add = TRUE, col = "blue", lty = 3)

hist(rBMT.Psi(10000), freq = FALSE, breaks = seq(-4, 4, 0.25), border = "blue")
curve(dlogis(x, scale = 1 / 1.70174439), add = TRUE, col = "red", lty = 2)
curve(dBMT.Psi(x), add = TRUE, col = "blue", lty = 3)

---

The BMT Distribution Descriptive Measures - Central Tendency.

Description

Mean, median and mode for the BMT distribution, with p3 and p4 tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

Usage

BMTmean(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

BMTmedian(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1,
  type.p.1.2 = "c-d")

BMTmode(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

Arguments

p3, p4	tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT distribution.
type.p.3.4	type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
p1, p2	domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution.
type.p.1.2	type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.

Details

See References.

Value

BMTmean gives the mean, BMTmedian the median and BMTmode the mode for the BMT distribution.

The arguments are recycled to the length of the result. Only the first elements of type.p.3.4 and type.p.1.2 are used.

If type.p.3.4 == "t w", p3 < 0 and p3 > 1 are errors and return NaN.

If type.p.3.4 == "a-s", p3 < -1 and p3 > 1 are errors and return NaN.

p4 < 0 and p4 > 1 are errors and return NaN.

If type.p.1.2 == "c-d", p1 >= p2 is an error and returns NaN.

If type.p.1.2 == "l-s", p2 <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

References

Torres-Jimenez, C. J. and Montenegro-Diaz, A. M. (2017, September), An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

BMTdispersion, BMTskewness, BMTkurtosis, BMTmoments for other descriptive measures or moments.

Examples

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTmean(0.25, 0.75, "t w")
BMTmedian(0.25, 0.75, "t w")
BMTmode(0.25, 0.75, "t w")

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTmean(0.5, 0.5, "a-s")
BMTmedian(0.5, 0.5, "a-s")
BMTmode(0.5, 0.5, "a-s")

# BMT on [-1.783489,3.312195] with 
# left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTmean(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTmedian(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTmode(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")

# BMT with mean equal to 0, standard deviation equal to 1, 
# asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTmean(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTmedian(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTmode(0.5, 0.5, "a-s", 0, 1, "l-s")

---

The BMT Distribution Parameter Conversion.

Description

Parameter conversion for different parameterizations for the BMT distribution, with p3 and p4 tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

Usage

BMTchangepars(p3, p4, type.p.3.4 = "t w", p1 = NULL, p2 = NULL,
  type.p.1.2 = NULL)

Arguments

p3, p4	tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT distribution.
type.p.3.4	type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
p1, p2	domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution.
type.p.1.2	type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.

Details

The BMT coefficient of asymmetry $-1 < \zeta < 1$ is

$\kappa_r - \kappa_l$

The BMT coefficient of steepness $0 < \xi < 1$ is

$(\kappa_r + \kappa_l - |\kappa_r - \kappa_l|) / (2 (1 - |\kappa_r - \kappa_l|))$

for $|\kappa_r - \kappa_l| < 1$.

The BMT distribution has mean $( d - c ) BMTmean(\kappa_l, \kappa_r) + c$ and standard deviation $( d - c ) BMTsd(\kappa_l, \kappa_r)$

From these equations, we can go back and forth with each parameterization.

Value

BMTchangepars reparametrizes p3, p4, p1, p2 according to the alternative parameterizations from the given type.p.3.4 and type.p.1.2. BMTchangepars returns a list with the alternative arguments to those received.

The arguments are recycled to the length of the result. Only the first elements of type.p.3.4 and type.p.1.2 are used.

If type.p.3.4 == "t w", p3 < 0 and p3 > 1 are errors and return NaN.

If type.p.3.4 == "a-s", p3 < -1 and p3 > 1 are errors and return NaN.

p4 < 0 and p4 > 1 are errors and return NaN.

If type.p.1.2 == "c-d", p1 >= p2 is an error and returns NaN.

If type.p.1.2 == "l-s", p2 <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected] and Alvaro Mauricio Montenegro Diaz [ths]

References

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

BMT for the BMT density, distribution, quantile function and random deviates.

Examples

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
parameters <- BMTchangepars(0.25, 0.75, "t w")
parameters # Parameters of the BMT in the asymmetry-steepness parametrization

# BMT with mean equal to 0, standard deviation equal to 1, 
# asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
parameters <- BMTchangepars(0.5, 0.5, "a-s", 0, 1, "l-s")
parameters # Parameters of the BMT in the tail weight and domain parametrization

---

The BMT Distribution Descriptive Measures - Dispersion.

Description

Variance, standard deviation and interquantile range for the BMT distribution, with p3 and p4 tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

Usage

BMTvar(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

BMTsd(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

BMTiqr(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

Arguments

p3, p4	tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT ditribution.
type.p.3.4	type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
p1, p2	domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution.
type.p.1.2	type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.

Details

See References.

Value

BMTvar gives the variance, BMTsd the standard deviation and BMTiqr the interquantile range for the BMT distribution.

The arguments are recycled to the length of the result. Only the first elements of type.p.3.4 and type.p.1.2 are used.

If type.p.3.4 == "t w", p3 < 0 and p3 > 1 are errors and return NaN.

If type.p.3.4 == "a-s", p3 < -1 and p3 > 1 are errors and return NaN.

p4 < 0 and p4 > 1 are errors and return NaN.

If type.p.1.2 == "c-d", p1 >= p2 is an error and returns NaN.

If type.p.1.2 == "l-s", p2 <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

References

Torres-Jimenez, C. J. and Montenegro-Diaz, A. M. (2017, September), An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

BMTcentral, BMTskewness, BMTkurtosis, BMTmoments for other descriptive measures or moments.

Examples

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTvar(0.25, 0.75, "t w")
BMTsd(0.25, 0.75, "t w")
BMTiqr(0.25, 0.75, "t w")

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTvar(0.5, 0.5, "a-s")
BMTsd(0.5, 0.5, "a-s")
BMTiqr(0.5, 0.5, "a-s")

# BMT on [-1.783489,3.312195] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTvar(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTsd(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTiqr(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")

# BMT with mean equal to 0, standard deviation equal to 1, 
# asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTvar(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTsd(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTiqr(0.5, 0.5, "a-s", 0, 1, "l-s")

---

Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by maximum likelihood (mle), moment matching (mme), quantile matching (qme), maximum goodness-of-fit (mge), also known as minimum distance, maximum product of spacing (mpse), also called maximum spacing, and minimum quantile distance (mqde), which can also be called maximum quantile goodness-of-fit.

Usage

BMTfit(data, method = c("mle", "mme", "qme", "mge", "mpse", "mqde"),
  start = list(p3 = 0.5, p4 = 0.5, p1 = min(data) - 0.1, p2 = max(data) +
  0.1), fix.arg = NULL, type.p.3.4 = "t w", type.p.1.2 = "c-d",
  optim.method = "Nelder-Mead", custom.optim = NULL, keepdata = TRUE,
  keepdata.nb = 100, ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
method	A character string coding for the fitting method: "mle" for 'maximum likelihood estimation', "mme" for 'moment matching estimation', "qme" for 'quantile matching estimation', "mge" for 'maximum goodness-of-fit estimation', "mpse" for 'maximum product of spacing estimation', and "mqde" for 'minimum quantile estimation'.
start	A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4	Type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2	Type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method	"default" (see the 'details' section of mledist) or optimization method to pass to optim.
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
keepdata	A logical. If TRUE, dataset is returned, otherwise only a sample subset is returned.
keepdata.nb	When keepdata=FALSE, the length (>1) of the subset returned.
...	Further arguments to be passed to generic functions, or to one of the functions "BMTfit.mle", "BMTfit.mme", "BMTfit.qme", "BMTfit.mge", "BMTfit.mpse", or "BMTfit.mqde" depending of the chosen method. See BMTfit.mle, BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mpse, BMTfit.mqde for details on parameter estimation.

Details

This function is based on the function fitdist from the package fitdistrplus but it focuses on the parameter estimation for the BMT distribution (see BMT for details). It has six possible fitting methods: maximum likelihood (mle), moment matching (mme), quantile matching (qme), maximum goodness-of-fit (mge), also known as minimum distance, maximum product of spacing (mpse), also called maximum spacing, and minimum quantile distance (mqde), which can also be called maximum quantile goodness-of-fit. These fitting methods are carried out in BMTfit.mle, BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mpse, and BMTfit.mqde, respectively (see each function for details). BMTfit returns an object of class "fitdist" (see fitdist for details). Therefore, it benefits of all the developed functions and methods for that class (see fitdistrplus for details).

Generic methods of a fitdist object are print, plot, summary, quantile, logLik, vcov and coef.

Value

fitdist returns an object of class "fitdist" with the following components:

estimate	the parameter estimates.
method	the character string coding for the fitting method : "mle" for 'maximum likelihood estimation', "mme" for 'moment matching estimation', "qme" for 'quantile matching estimation', "mge" for 'maximum goodness-of-fit estimation', "mpse" for 'maximum product of spacing estimation', and "mqde" for 'minimum quantile estimation'.
sd	the estimated standard errors, NA if numerically not computable or NULL if not available.
cor	the estimated correlation matrix, NA if numerically not computable or NULL if not available.
vcov	the estimated variance-covariance matrix, NULL if not available.
loglik	the log-likelihood.
aic	the Akaike information criterion.
bic	the the so-called BIC or SBC (Schwarz Bayesian criterion).
n	the length of the data set.
data	the data set.
distname	the name of the distribution (BMT).
fix.arg	the named list giving the values of parameters of the named distribution that must be kept fixed rather than estimated or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
discrete	the input argument or the automatic definition by the function to be passed to functions gofstat, plotdist and cdfcomp.
dots	the list of further arguments passed in ... to be used in bootdist in iterative calls to mledist, mmedist, qmedist, mgedist, mpsedist, mqdedist or NULL if no such arguments.
weights	the vector of weigths used in the estimation process or NULL.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function fitdist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

References

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mle, BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mpse and BMTfit.mqde for details on parameter estimation. See fitdist for details on the object fitdist and its methods print, plot, summary, quantile, logLik, vcov and coef, and fitdistrplus for an overview of the package to which that object belongs to.

Examples

# (1) fit of the BMT distribution by maximum likelihood estimation
data(groundbeef)
serving <- groundbeef$serving
fit.mle <- BMTfit(serving)
summary(fit.mle)
plot(fit.mle)
plot(fit.mle, demp = TRUE)
plot(fit.mle, histo = FALSE, demp = TRUE)
cdfcomp(fit.mle, addlegend=FALSE)
denscomp(fit.mle, addlegend=FALSE)
ppcomp(fit.mle, addlegend=FALSE)
qqcomp(fit.mle, addlegend=FALSE)

# (2) Comparison of various estimation methods
fit.mme <- BMTfit(serving, method="mme")
fit.mpse <- BMTfit(serving, method="mpse")
fit.mqde <- BMTfit(serving, method="mqde")
summary(fit.mme)
summary(fit.mpse)
summary(fit.mqde)
cdfcomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), 
        legendtext=c("mle", "mme", "mpse", "mqde"))
denscomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), 
         legendtext=c("mle", "mme", "mpse", "mqde"))
qqcomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), 
       legendtext=c("mle", "mme", "mpse", "mqde"))
ppcomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), 
       legendtext=c("mle", "mme", "mpse", "mqde"))
gofstat(list(fit.mle, fit.mme, fit.mpse, fit.mqde), 
        fitnames=c("mle", "mme", "mpse", "mqde"))

# (3) how to change the optimisation method?
BMTfit(serving, optim.method="Nelder-Mead")
BMTfit(serving, optim.method="L-BFGS-B") 
BMTfit(serving, custom.optim="nlminb")

# (4) estimation of the tails weights parameters of the BMT distribution 
# with domain fixed at [9,201] using Kolmogorov-Smirnov
fit.KS <- BMTfit(serving, method="mge", gof="KS", 
                 start=list(p3=0.5, p4=0.5), fix.arg=list(p1=9, p2=201))
summary(fit.KS)
plot(fit.KS)

# (5) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [9,201] using minimum quantile distance 
# with a closed formula (optim.method="CD")
fit.mqde.CD <- BMTfit(serving, method="mqde", optim.method="CD", 
                      start=list(p3=0.5, p4=0.5), type.p.3.4 = "a-s", 
                      fix.arg=list(p1=9, p2=201))
summary(fit.mqde.CD)
plot(fit.mqde.CD)

---

Maximum Goodness-of-fit Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by maximum goodness-of-fit estimation (mge), also known as minimum distance estimation.

Usage

BMTfit.mge(data, gof = "CvM", start = list(p3 = 0.5, p4 = 0.5, p1 =
  min(data) - 0.1, p2 = max(data) + 0.1), fix.arg = NULL,
  type.p.3.4 = "t w", type.p.1.2 = "c-d", optim.method = "Nelder-Mead",
  custom.optim = NULL, silent = TRUE, ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
gof	A character string coding for the name of the goodness-of-fit distance used : "CvM" for Cramer-von Mises distance,"KS" for Kolmogorov-Smirnov distance, "AD" for Anderson-Darling distance, "ADR", "ADL", "AD2R", "AD2L" and "AD2" for variants of Anderson-Darling distance described by Luceno (2006).
start	A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4	Type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2	Type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method	"default" (see the 'details' section of mledist) or optimization method to pass to optim.
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
silent	A logical to remove or show warnings when bootstraping.
...	Further arguments to be passed to generic functions or to the function "mgedist". See mgedist for details.

Details

This function is not intended to be called directly but is internally called in BMTfit when used with the maximum goodness-of-fit method.

BMTfit.mge is based on the function mgedist but it focuses on the maximum goodness-of-fit parameter estimation for the BMT distribution (see BMT for details about the BMT distribution and mgedist for details about maximum goodness-of-fit fit of univariate distributions).

Value

BMTfit.mge returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value	the value of the corresponding objective function of the estimation method at the estimate.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function.
loglik	the log-likelihood value.
gof	the code of the goodness-of-fit distance maximized.
optim.function	the name of the optimization function used for maximum product of spacing.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied function or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function mgedist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

References

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mme, BMTfit.qme, BMTfit.mle, BMTfit.mpse and BMTfit.mqde for other estimation methods. See optim and constrOptim for optimization routines. See BMTfit and fitdist for functions that return an objetc of class "fitdist".

Examples

# (1) basic fit by maximum goodness-of-fit estimation
set.seed(1234)
x1 <- rBMT(n=100, p3 = 0.25, p4 = 0.75)
BMTfit.mge(x1)

# (2) how to change the goodness-of-fit statistic/distance?
BMTfit.mge(x1, gof="KS")
BMTfit.mge(x1, gof="AD2R")

# (3) how to change the optimisation method?
BMTfit.mge(x1, optim.method="L-BFGS-B") 
BMTfit.mge(x1, custom.optim="nlminb")

# (4) estimation of the tails weights parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mge(x1, start=list(p3=0.5, p4=0.5), fix.arg=list(p1=0, p2=1))

# (5) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mge(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
           fix.arg=list(p1=0, p2=1))

---

Maximum Likelihood Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by maximum likelihood estimation (mle).

Usage

BMTfit.mle(data, start = list(p3 = 0.5, p4 = 0.5, p1 = min(data) - 0.1, p2 =
  max(data) + 0.1), fix.arg = NULL, type.p.3.4 = "t w",
  type.p.1.2 = "c-d", optim.method = "Nelder-Mead", custom.optim = NULL,
  silent = TRUE, ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
start	A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4	Type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2	Type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method	"default" (see the 'details' section of mledist) or optimization method to pass to optim.
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
silent	A logical to remove or show warnings when bootstraping.
...	Further arguments to be passed to generic functions or to the function "mledist". See mledist for details.

Details

This function is not intended to be called directly but is internally called in BMTfit when used with the maximum likelihood method.

BMTfit.mle is based on the function mledist from the package fitdistrplus but it focuses on the maximum likelihood parameter estimation for the BMT distribution (see BMT for details about the BMT distribution and mledist for details about maximum likelihood fit of univariate distributions).

Value

BMTfit.mle returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
loglik	the log-likelihood value.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function. It is used in BMTfit to estimate standard errors.
optim.function	the name of the optimization function used for maximum likelihood.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied function or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function mledist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

References

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mpse and BMTfit.mqde for other estimation methods. See optim and constrOptim for optimization routines. See BMTfit and fitdist for functions that return an objetc of class "fitdist".

Examples

# (1) basic fit by maximum likelihood estimation
set.seed(1234)
x1 <- rBMT(n=100, p3 = 0.25, p4 = 0.75)
BMTfit.mle(x1)

# (2) how to change the optimisation method?
BMTfit.mle(x1, optim.method="L-BFGS-B") 
BMTfit.mle(x1, custom.optim="nlminb")

# (3) estimation of the tails weights parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mle(x1, start=list(p3=0.5, p4=0.5), fix.arg=list(p1=0, p2=1))

# (4) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mle(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
           fix.arg=list(p1=0, p2=1))

---

Moment Matching Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by moment matching (mme).

Usage

BMTfit.mme(data, start = list(p3 = 0.5, p4 = 0.5, p1 = min(data) - 0.1, p2 =
  max(data) + 0.1), fix.arg = NULL, type.p.3.4 = "t w",
  type.p.1.2 = "c-d", optim.method = "Nelder-Mead", custom.optim = NULL,
  silent = TRUE, ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
start	A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4	Type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2	Type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method	"default" (see the 'details' section of mledist) or optimization method to pass to optim.
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
silent	A logical to remove or show warnings when bootstraping.
...	Further arguments to be passed to generic functions or to the function "mmedist". See mmedist for details.

Details

This function is not intended to be called directly but is internally called in BMTfit when used with the moment matching method.

BMTfit.mme is based on the function mmedist but it focuses on the moment matching parameter estimation for the BMT distribution (see BMT for details about the BMT distribution and mmedist for details about moment matching fit of univariate distributions).

For each parameter of the BMT distribution we choose a moment or measure. Mean for p1, standard deviation for p2, Pearson_s skewness for p3, and Pearson's kurtosis for p4.

Value

BMTfit.mme returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value	the value of the corresponding objective function of the estimation method at the estimate.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function.
loglik	the log-likelihood value.
order	the vector of moment(s) matched: mean (1), standard deviation (2), Pearson's skewness (3), Pearson's kurtosis (4).
memp	the empirical moment function.
optim.function	the name of the optimization function used for maximum product of spacing.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied function or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function mmedist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

References

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.qme, BMTfit.mle, BMTfit.mge, BMTfit.mpse and BMTfit.mqde for other estimation methods. See optim and constrOptim for optimization routines. See BMTfit and fitdist for functions that return an objetc of class "fitdist".

Examples

# (1) basic fit by moment matching estimation
set.seed(1234)
x1 <- rBMT(n=100, p3=0.25, p4=0.75)
BMTfit.mme(x1)

# (3) how to change the optimisation method?
BMTfit.mme(x1, optim.method="L-BFGS-B") 
BMTfit.mme(x1, custom.optim="nlminb")

# (4) estimation of the tails weights parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mme(x1, start=list(p3=0.5, p4=0.5), fix.arg=list(p1=0, p2=1))

# (5) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mme(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
           fix.arg=list(p1=0, p2=1))

---

Maximum Product of Spacing Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by maximum product of spacing estimation (mpse), also called maximum spacing estimation.

Usage

BMTfit.mpse(data, start = list(p3 = 0.5, p4 = 0.5, p1 = min(data) - 0.1, p2 =
  max(data) + 0.1), fix.arg = NULL, type.p.3.4 = "t w",
  type.p.1.2 = "c-d", optim.method = "Nelder-Mead", custom.optim = NULL,
  silent = TRUE, ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
start	A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4	Type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2	Type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method	"default" (see the 'details' section of mledist) or optimization method to pass to optim.
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
silent	A logical to remove or show warnings when bootstraping.
...	Further arguments to be passed to generic functions or to the function "mpsedist". See mpsedist for details.

Details

This function is not intended to be called directly but is internally called in BMTfit when used with the maximum product of spacing method.

BMTfit.mpse is based on the function mpsedist but it focuses on the maximum product of spacing parameter estimation for the BMT distribution (see BMT for details about the BMT distribution and mpsedist for details about maximum product of spacing fit of univariate distributions).

Value

BMTfit.mpse returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value	the value of the corresponding objective function of the estimation method at the estimate.
loglik	the log-likelihood value.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function.
optim.function	the name of the optimization function used for maximum product of spacing.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied function or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function mpsedist which in turn is based on the function mledist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

References

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mle and BMTfit.mqde for other estimation methods. See optim and constrOptim for optimization routines. See BMTfit and fitdist for functions that return an objetc of class "fitdist".

Examples

# (1) basic fit by maximum product of spacing estimation
set.seed(1234)
x1 <- rBMT(n=100, p3 = 0.25, p4 = 0.75)
BMTfit.mpse(x1)

# (2) how to change the optimisation method?
BMTfit.mpse(x1, optim.method="L-BFGS-B") 
BMTfit.mpse(x1, custom.optim="nlminb")

# (3) estimation of the tails weights parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mpse(x1, start=list(p3=0.5, p4=0.5), fix.arg=list(p1=0, p2=1))

# (4) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mpse(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
            fix.arg=list(p1=0, p2=1))

---

Minimum Quantile Distance Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by minimum quantile distance (mqde), which can also be called maximum quantile goodness-of-fit.

Usage

BMTfit.mqde(data, probs = (1:length(data) - 0.5)/length(data), qtype = 5,
  dist = "euclidean", start = list(p3 = 0.5, p4 = 0.5, p1 = min(data) - 0.1,
  p2 = max(data) + 0.1), fix.arg = NULL, type.p.3.4 = "t w",
  type.p.1.2 = "c-d", optim.method = "Nelder-Mead", custom.optim = NULL,
  weights = NULL, silent = TRUE, ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
probs	A numeric vector of the probabilities for which the minimum quantile distance estimation is done. $p[k] = (k - 0.5) / n$ (default).
qtype	The quantile type used by the R quantile function to compute the empirical quantiles. Type 5 (default), i.e. $x[k]$ is both the $k$th order statistic and the type 5 sample quantile of $p[k] = (k - 0.5) / n$.
dist	The distance measure between observed and theoretical quantiles to be used. This must be one of "euclidean" (default), "maximum", or "manhattan". Any unambiguous substring can be given.
start	A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4	Type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2	Type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method	"default" (see the 'details' section of mledist) or optimization method to pass to optim. Given the close-form expression of the quantile function, two optimization methods were added when the euclidean distance is selected: Coordinate descend ("CD") and Newton-Rhapson ("NR").
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
weights	an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector with strictly positive numbers. If non-NULL, weighted mqde is used, otherwise ordinary mqde.
silent	A logical to remove or show warnings when bootstraping.
...	Further arguments to be passed to generic functions or to the function "mqdedist". See mqdedist for details.

Details

This function is not intended to be called directly but is internally called in BMTfit when used with the minimum quantile distance method.

BMTfit.mqde is based on the function mqdedist but it focuses on the minimum quantile distance parameter estimation for the BMT distribution (see BMT for details about the BMT distribution and mqdedist for details about minimum quantile distance fit of univariate distributions).

Given the close-form expression of the quantile function, two optimization methods were added when the euclidean distance is selected: Coordinate descend ("CD") and Newton-Rhapson ("NR").

Value

BMTfit.mqde returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value	the value of the corresponding objective function of the estimation method at the estimate.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function.
loglik	the log-likelihood value.
probs	the probability vector on which observed and theoretical quantiles were calculated.
dist	the name of the distance between observed and theoretical quantiles used.
optim.function	the name of the optimization function used for maximum product of spacing.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied function or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function mqdedist which in turn is based on the function mledist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

References

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mme, BMTfit.mle, BMTfit.mge, BMTfit.mpse and BMTfit.qme for other estimation methods. See optim and constrOptim for optimization routines. See BMTfit and fitdist for functions that return an objetc of class "fitdist".

Examples

# (1) basic fit by minimum quantile distance estimation
set.seed(1234)
x1 <- rBMT(n=100, p3=0.25, p4=0.75)
BMTfit.mqde(x1)

# (2) quantile matching is a particular case of minimum quantile distance
BMTfit.mqde(x1, probs=c(0.2,0.4,0.6,0.8), qtype=7)

# (3) maximum or manhattan instead of euclidean distance
BMTfit.mqde(x1, dist="maximum")
BMTfit.mqde(x1, dist="manhattan")

# (4) how to change the optimisation method?
BMTfit.mqde(x1, optim.method="L-BFGS-B") 
BMTfit.mqde(x1, custom.optim="nlminb")

# (5) estimation of the tails weights parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mqde(x1, start=list(p3=0.5, p4=0.5), fix.arg=list(p1=0, p2=1))

# (6) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mqde(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
            fix.arg=list(p1=0, p2=1))

---

Quantile Matching Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by quantile matching estimation (qme).

Usage

BMTfit.qme(data, probs = c(0.2, 0.4, 0.6, 0.8), qtype = 7, start = list(p3
  = 0.5, p4 = 0.5, p1 = min(data) - 0.1, p2 = max(data) + 0.1),
  fix.arg = NULL, type.p.3.4 = "t w", type.p.1.2 = "c-d",
  optim.method = "Nelder-Mead", custom.optim = NULL, silent = TRUE, ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
probs	A numeric vector of the probabilities for which the quantile matching is done. The length of this vector must be equal to the number of parameters to estimate.
qtype	The quantile type used by the R quantile function to compute the empirical quantiles, (default 7 corresponds to the default quantile method in R).
start	A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4	Type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2	Type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method	"default" (see the 'details' section of mledist) or optimization method to pass to optim.
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
silent	A logical to remove or show warnings when bootstraping.
...	Further arguments to be passed to generic functions or to the function "qmedist". See qmedist for details.

Details

This function is not intended to be called directly but is internally called in BMTfit when used with the quantile matching method.

BMTfit.qme is based on the function qmedist but it focuses on the quantile matching parameter estimation for the BMT distribution (see BMT for details about the BMT distribution and qmedist for details about quantile matching fit of univariate distributions).

Value

BMTfit.qme returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value	the value of the corresponding objective function of the estimation method at the estimate.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function.
loglik	the log-likelihood value.
probs	the probability vector on which quantiles are matched.
optim.function	the name of the optimization function used for maximum product of spacing.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied function or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function qmedist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

References

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mme, BMTfit.mle, BMTfit.mge, BMTfit.mpse and BMTfit.mqde for other estimation methods. See optim and constrOptim for optimization routines. See BMTfit and fitdist for functions that return an objetc of class "fitdist".

Examples

# (1) basic fit by quantile matching estimation
set.seed(1234)
x1 <- rBMT(n=100, p3 = 0.25, p4 = 0.75)
BMTfit.qme(x1)

# (2) changing the probability vector on which quantiles are matched
BMTfit.qme(x1, probs=c(0.1,0.3,0.5,0.75))

# (3) how to change the optimisation method?
BMTfit.qme(x1, optim.method="L-BFGS-B") 
BMTfit.qme(x1, custom.optim="nlminb")

# (4) estimation of the tails weights parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.qme(x1, start=list(p3=0.5, p4=0.5), 
           fix.arg=list(p1=0, p2=1), probs=c(1/3,2/3))

# (5) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.qme(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
           fix.arg=list(p1=0, p2=1), probs=c(1/3,2/3))

---

The BMT Distribution Descriptive Measures - Kurtosis.

Description

Kurtosis and steepness coefficient for the BMT distribution with p3 and p4 tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

Usage

BMTkurt(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

BMTsteep(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

Arguments

p3, p4	tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT distribution.
type.p.3.4	type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
p1, p2	domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution.
type.p.1.2	type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.

Details

See References.

Value

BMTkurt gives the Pearson's kurtosis and BMTsteep the proposed steepness coefficient for the BMT distribution.

The arguments are recycled to the length of the result. Only the first elements of type.p.3.4 and type.p.1.2 are used.

If type.p.3.4 == "t w", p3 < 0 and p3 > 1 are errors and return NaN.

If type.p.3.4 == "a-s", p3 < -1 and p3 > 1 are errors and return NaN.

p4 < 0 and p4 > 1 are errors and return NaN.

If type.p.1.2 == "c-d", p1 >= p2 is an error and returns NaN.

If type.p.1.2 == "l-s", p2 <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

References

Torres-Jimenez, C. J. and Montenegro-Diaz, A. M. (2017, September), An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

BMTcentral, BMTdispersion, BMTskewness, BMTmoments for other descriptive measures or moments.

Examples

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTkurt(0.25, 0.75, "t w")
BMTsteep(0.25, 0.75, "t w")

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTkurt(0.5, 0.5, "a-s")
BMTsteep(0.5, 0.5, "a-s")

# domain or location-scale parameters do not affect 
# the skewness and the asymmetry coefficient

# BMT on [-1.783489,3.312195] with 
# left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTkurt(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTsteep(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")

# BMT with mean equal to 0, standard deviation equal to 1, 
# asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTkurt(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTsteep(0.5, 0.5, "a-s", 0, 1, "l-s")

---

The BMT Distribution Moments, Moment-Generating Function and Characteristic Function.

Description

Any raw, central or standarised moment, the moment-generating function and the characteristic function for the BMT distribution, with p3 and p4 tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

Usage

BMTmoment(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1,
  type.p.1.2 = "c-d", order, type = "standardised", method = "quadrature")

BMTmgf(s, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1,
  type.p.1.2 = "c-d")

BMTchf(s, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1,
  type.p.1.2 = "c-d")

mBMT(order, p3, p4, type.p.3.4, p1, p2, type.p.1.2)

Arguments

p3, p4	tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT distribution.
type.p.3.4	type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
p1, p2	domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution.
type.p.1.2	type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
order	order of the moment.
type	type of the moment: raw, central or standardised (default).
method	method to obtain the moment: exact formula or Chebyshev-Gauss quadrature (default).
s	variable for the moment-generating and characteristic functions.

Details

See References.

Value

BMTmoment gives any raw, central or standarised moment, BMTmgf the moment-generating function and BMTchf the characteristic function

The arguments are recycled to the length of the result. Only the first elements of type.p.3.4, type.p.1.2, type and method are used.

If type.p.3.4 == "t w", p3 < 0 and p3 > 1 are errors and return NaN.

If type.p.3.4 == "a-s", p3 < -1 and p3 > 1 are errors and return NaN.

p4 < 0 and p4 > 1 are errors and return NaN.

If type.p.1.2 == "c-d", p1 >= p2 is an error and returns NaN.

If type.p.1.2 == "l-s", p2 <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

References

Torres-Jimenez, C. J. and Montenegro-Diaz, A. M. (2017, September), An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

BMTcentral, BMTdispersion, BMTskewness, BMTkurtosis for specific descriptive measures or moments.

Examples

layout(matrix(1:4, 2, 2, TRUE))
s <- seq(-1, 1, length.out = 100)

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTmoment(0.25, 0.75, order = 5) # hyperskewness by Gauss-Legendre quadrature
BMTmoment(0.25, 0.75, order = 5, method = "exact") # hyperskewness by exact formula
mgf <- BMTmgf(s, 0.25, 0.75) # moment-generation function
plot(s, mgf, type="l")
chf <- BMTchf(s, 0.25, 0.75) # characteristic function

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.5
BMTmoment(0.5, 0.5, "a-s", order = 5)
BMTmoment(0.5, 0.5, "a-s", order = 5, method = "exact")
mgf <- BMTmgf(s, 0.5, 0.5, "a-s")
plot(s, mgf, type="l")
chf <- BMTchf(s, 0.5, 0.5, "a-s")

# BMT on [-1.783489, 3.312195] with 
# left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTmoment(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d", order = 5)
BMTmoment(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d", order = 5, method = "exact")
mgf <- BMTmgf(s, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
plot(s, mgf, type="l")
chf <- BMTchf(s, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")

# BMT with mean equal to 0, standard deviation equal to 1, 
# asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.5
BMTmoment(0.5, 0.5, "a-s", 0, 1, "l-s", order = 5)
BMTmoment(0.5, 0.5, "a-s", 0, 1, "l-s", order = 5, method = "exact")
mgf <- BMTmgf(s, 0.5, 0.5, "a-s", 0, 1, "l-s")
plot(s, mgf, type="l")
chf <- BMTchf(s, 0.5, 0.5, "a-s", 0, 1, "l-s")

---

The BMT Distribution Descriptive Measures - Skewness.

Description

Skewness and an asymmetry coefficient for the BMT distribution, with p3 and p4 tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

Usage

BMTskew(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

BMTasymm(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

Arguments

p3, p4	tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT distribution.
type.p.3.4	type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
p1, p2	domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution.
type.p.1.2	type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.

Details

See References.

Value

BMTskew gives the Pearson's skewness and BMTasymm the proposed asymmetry coefficient for the BMT distribution.

The arguments are recycled to the length of the result. Only the first elements of type.p.3.4 and type.p.1.2 are used.

If type.p.3.4 == "t w", p3 < 0 and p3 > 1 are errors and return NaN.

If type.p.3.4 == "a-s", p3 < -1 and p3 > 1 are errors and return NaN.

p4 < 0 and p4 > 1 are errors and return NaN.

If type.p.1.2 == "c-d", p1 >= p2 is an error and returns NaN.

If type.p.1.2 == "l-s", p2 <= 0 is an error and returns NaN.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

References

Torres-Jimenez, C. J. and Montenegro-Diaz, A. M. (2017, September), An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

See Also

BMTcentral, BMTdispersion, BMTkurtosis, BMTmoments for other descriptive measures or moments.

Examples

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTskew(0.25, 0.75, "t w")
BMTasymm(0.25, 0.75, "t w")

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTskew(0.5, 0.5, "a-s")
BMTasymm(0.5, 0.5, "a-s")

# domain or location-scale parameters do not affect 
# the skewness and the asymmetry coefficient

# BMT on [-1.783489,3.312195] with 
# left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTskew(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTasymm(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")

# BMT with mean equal to 0, standard deviation equal to 1, 
# asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTskew(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTasymm(0.5, 0.5, "a-s", 0, 1, "l-s")

---

Maximum Product of Spacing Fit of Univariate Distributions.

Description

Fit of univariate distributions for non-censored data using maximum product of spacing estimation (mpse), also called maximum spacing estimation.

Usage

mpsedist(data, distr, start = NULL, fix.arg = NULL,
  optim.method = "default", lower = -Inf, upper = Inf,
  custom.optim = NULL, weights = NULL, silent = TRUE, gradient = NULL,
  ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
distr	A character string "name" naming a distribution for which the corresponding density function dname and the corresponding distribution function pname must be classically defined.
start	A named list giving the initial values of parameters of the named distribution or a function of data computing initial values and returning a named list. This argument may be omitted (default) for some distributions for which reasonable starting values are computed (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the named distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated.
optim.method	"default" (see details) or an optimization method to pass to optim.
lower	Left bounds on the parameters for the "L-BFGS-B" method (see optim) or the constrOptim function (as an equivalent linear constraint).
upper	Right bounds on the parameters for the "L-BFGS-B" method (see optim) or the constrOptim function (as an equivalent linear constraint).
custom.optim	A function carrying the optimization (see details).
weights	An optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector with strictly positive numbers. If non-NULL, weighted mpse is used, otherwise ordinary mpse.
silent	A logical to remove or show warnings when bootstraping.
gradient	A function to return the gradient of the optimization objective function for the "BFGS", "CG" and "L-BFGS-B" methods. If it is NULL, a finite-difference approximation will be used, see optim.
...	Further arguments passed to the optim, constrOptim or custom.optim function.

Details

The mpsedist function carries out the maximum product of spacing estimation numerically, by maximization of the arithmetic mean of $\log(F(k) - F(k-1))$.

The optimization process is the same as mledist, see the 'details' section of that function.

Optionally, a vector of weights can be used in the fitting process. By default (when weigths=NULL), ordinary mpse is carried out, otherwise the specified weights are used to compute a weighted arithmetic mean.

We believe this function should be added to the package fitdistrplus. Until it is accepted and incorporated into that package, it will remain in the package BMT. This function is internally called in BMTfit.mpse.

Value

mpsedist returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value	the value of the optimization objective function at the solution found.
loglik	the log-likelihood.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function.
optim.function	the name of the optimization function used.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated by maximum likelihood or NULL if there are no such parameters.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied optimization function, or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function mledist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

Functions checkparam and start.arg.default are needed and were copied from the same package (fitdistrplus version: 1.0-9).

References

Cheng, R. and N. Amin (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society. Series B (Methodological), 394-403.

Ranneby, B. (1984). The maximum spacing method. an estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics, 93-112.

See Also

mqdedist, mledist, mmedist, qmedist, mgedist, and optim.

Examples

# (1) basic fit of a normal distribution 
set.seed(1234)
x1 <- rnorm(n = 100)
mean(x1); sd(x1)
mpse1 <- mpsedist(x1, "norm")
mpse1$estimate

# (2) defining your own distribution functions, here for the Gumbel 
# distribution for other distributions, see the CRAN task view dedicated 
# to probability distributions
dgumbel <- function(x, a, b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b))
pgumbel <- function(q, a, b) exp(-exp((a-q)/b))
qgumbel <- function(p, a, b) a-b*log(-log(p))
mpse1 <- mpsedist(x1, "gumbel", start = list(a = 10, b = 5))
mpse1$estimate

# (3) fit a discrete distribution (Poisson)
set.seed(1234)
x2 <- rpois(n = 30, lambda = 2)
mpse2 <- mpsedist(x2, "pois")
mpse2$estimate

# (4) fit a finite-support distribution (beta)
set.seed(1234)
x3 <- rbeta(n = 100, shape1 = 5, shape2 = 10)
mpse3 <- mpsedist(x3, "beta")
mpse3$estimate

# (5) fit frequency distributions on USArrests dataset.
x4 <- USArrests$Assault
mpse4pois <- mpsedist(x4, "pois")
mpse4pois$estimate
mpse4nbinom <- mpsedist(x4, "nbinom")
mpse4nbinom$estimate

# (6) weighted fit of a normal distribution 
set.seed(1234)
w1 <- runif(101)
mpse1 <- mpsedist(x1, "norm", weights = w1)
mpse1$estimate

---

Minimum Quantile Distance Fit of Univariate Distributions.

Description

Fit of univariate distributions for non-censored data using minimum quantile distance estimation (mqde), which can also be called maximum quantile goodness-of-fit estimation.

Usage

mqdedist(data, distr, probs = (1:length(data) - 0.5)/length(data),
  qtype = 5, dist = "euclidean", start = NULL, fix.arg = NULL,
  optim.method = "default", lower = -Inf, upper = Inf,
  custom.optim = NULL, weights = NULL, silent = TRUE, gradient = NULL,
  ...)

Arguments

data	A numeric vector with the observed values for non-censored data.
distr	A character string "name" naming a distribution for which the corresponding quantile function qname and the corresponding density distribution dname must be classically defined.
probs	A numeric vector of the probabilities for which the minimum quantile distance estimation is done. $p[k] = (k - 0.5) / n$ (default).
qtype	The quantile type used by the R quantile function to compute the empirical quantiles. Type 5 (default), i.e. $x[k]$ is both the $k$th order statistic and the type 5 sample quantile of $p[k] = (k - 0.5) / n$.
dist	The distance measure between observed and theoretical quantiles to be used. This must be one of "euclidean" (default), "maximum", or "manhattan". Any unambiguous substring can be given.
start	A named list giving the initial values of parameters of the named distribution or a function of data computing initial values and returning a named list. This argument may be omitted (default) for some distributions for which reasonable starting values are computed (see the 'details' section of mledist).
fix.arg	An optional named list giving the values of fixed parameters of the named distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated.
optim.method	"default" (see details) or optimization method to pass to optim.
lower	Left bounds on the parameters for the "L-BFGS-B" method (see optim) or the constrOptim function (as an equivalent linear constraint).
upper	Right bounds on the parameters for the "L-BFGS-B" method (see optim) or the constrOptim function (as an equivalent linear constraint).
custom.optim	A function carrying the optimization (see details).
weights	An optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector with strictly positive numbers. If non-NULL, weighted mqde is used, otherwise ordinary mqde.
silent	A logical to remove or show warnings when bootstraping.
gradient	A function to return the gradient of the optimization objective function for the "BFGS", "CG" and "L-BFGS-B" methods. If it is NULL, a finite-difference approximation will be used, see optim.
...	Further arguments passed to the optim, constrOptim or custom.optim function.

Details

The mqdedist function carries out the minimum quantile distance estimation numerically, by minimization of a distance between observed and theoretical quantiles.

The optimization process is the same as mledist, see the 'details' section of that function.

Optionally, a vector of weights can be used in the fitting process. By default (when weigths=NULL), ordinary mqde is carried out, otherwise the specified weights are used to compute a weighted distance.

We believe this function should be added to the package fitdistrplus. Until it is accepted and incorporated into that package, it will remain in the package BMT. This function is internally called in BMTfit.mqde.

Value

mqdedist returns a list with following components,

estimate	the parameter estimates.
convergence	an integer code for the convergence of optim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value	the value of the optimization objective function at the solution found.
hessian	a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function.
probs	the probability vector on which observed and theoretical quantiles were calculated.
dist	the name of the distance between observed and theoretical quantiles used.
optim.function	the name of the optimization function used.
fix.arg	the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated by maximum likelihood or NULL if there are no such parameters.
loglik	the log-likelihood.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
weights	the vector of weigths used in the estimation process or NULL.
counts	A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied optimization function, or set to NULL.
optim.message	A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

Based on the function mledist of the R package: fitdistrplus

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

Functions checkparam and start.arg.default are needed and were copied from the same package (fitdistrplus version: 1.0-9).

References

LaRiccia, V. N. (1982). Asymptotic Properties of Weighted $L^2$ Quantile Distance Estimators. The Annals of Statistics, 10(2), 621-624.

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

See Also

mpsedist, mledist, mmedist, qmedist, mgedist, optim, constrOptim, and quantile.

Examples

# (1) basic fit of a normal distribution 
set.seed(1234)
x1 <- rnorm(n = 100)
mean(x1); sd(x1)
mqde1 <- mqdedist(x1, "norm")
mqde1$estimate

# (2) defining your own distribution functions, here for the Gumbel 
# distribution for other distributions, see the CRAN task view dedicated 
# to probability distributions
dgumbel <- function(x, a, b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b))
pgumbel <- function(q, a, b) exp(-exp((a-q)/b))
qgumbel <- function(p, a, b) a-b*log(-log(p))
mqde1 <- mqdedist(x1, "gumbel", start = list(a = 10, b = 5))
mqde1$estimate

# (3) fit a discrete distribution (Poisson)
set.seed(1234)
x2 <- rpois(n = 30, lambda = 2)
mqde2 <- mqdedist(x2, "pois")
mqde2$estimate

# (4) fit a finite-support distribution (beta)
set.seed(1234)
x3 <- rbeta(n = 100, shape1 = 5, shape2 = 10)
mqde3 <- mqdedist(x3, "beta")
mqde3$estimate

# (5) fit frequency distributions on USArrests dataset.
x4 <- USArrests$Assault
mqde4pois <- mqdedist(x4, "pois")
mqde4pois$estimate
mqde4nbinom <- mqdedist(x4, "nbinom")
mqde4nbinom$estimate

# (6) weighted fit of a normal distribution 
set.seed(1234)
w1 <- runif(100)
weighted.mean(x1, w1)
mqde1 <- mqdedist(x1, "norm", weights = w1)
mqde1$estimate

---

A Performance Score in Mathematics from Booklet 10 of PISA 2012.

Description

Classic performance score in mathematics, using the students responses to Booklet 10 of the PISA test applied in 2012. We use the answers to the PISA 2012 questionnaire. We take the “Scored cognitive item response data file.”. We keep all the questions with binary score (correct and incorrect) and exclude those that could be scored with partial credit. Then, we recode the responses: 1 for correct and 0 for incorrect. Finally, we obtain the percentage of correct answers, i.e., the classic performance score of each student. It is important to mention that the reported scores of PISA 2012 are estimated and scaled using the Rasch model of item response theory. On the other hand, all booklets for the test could have different: number of questions, traits evaluated, and participating countries. Also, the assignation of a booklet to a student is randomized. Considering that, we choose only one arbitrary booklet, Booklet 10, and its questions of mathematics.

Usage

data(score.math.booklet10.PISA2012)

Format

score.math.booklet10.PISA2012 is a vector.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

Source

OECD Programme for International Student Assesment (PISA) (2012). Database - PISA 2012. Online; accessed 2014-08-23. https://www.oecd.org/pisa/pisaproducts/pisa2012database-downloadabledata.htm.

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