gen.betaIImr: Generalized Beta Distribution of the Second Kind family...
In VGAMextra: Additions and Extensions of the 'VGAM' Package
gen.betaIImr R Documentation
Generalized Beta Distribution of the Second Kind family function
Description
Maximum likelihood estimation of the 4-parameter
generalized beta II distribution using Fisher scoring.
Usage
gen.betaIImr(lscale = "loglink",
lshape1.a = "loglink",
lshape2.p = "loglink",
lshape3.q = "loglink",
iscale = NULL,
ishape1.a = NULL,
ishape2.p = NULL,
ishape3.q = NULL,
imethod = 1,
lss = TRUE,
gscale = exp(-5:5),
gshape1.a = exp(-5:5),
gshape2.p = exp(-5:5),
gshape3.q = exp(-5:5),
probs.y = c(0.25, 0.50, 0.75),
zero = "shape" )
Arguments
lscale, lshape1.a, lshape2.p, lshape3.q
Parameter link functions applied to the
shape parameter a,
scale parameter scale,
shape parameter p, and
shape parameter q.
All four parameters are positive.
See Links for more choices.
iscale, ishape1.a, ishape2.p, ishape3.q
Optional initial values for b, a, p and q.
Default is NULL for all of them, meaning initial values are
computed internally.
imethod
Initializing method to internally compute the initial values.
Currently, only method = 1 is handled.
gscale, gshape1.a, gshape2.p, gshape3.q
Grid search initial values.
See CommonVGAMffArguments for further information.
zero
Numeric or Character vector. Position(s) or name(s) of the
parameters/linear predictors to be modeled as intercept-only.
Details at
CommonVGAMffArguments.
lss, probs.y
See CommonVGAMffArguments for important information.
Details
This distribution is most useful for unifying a substantial
number of size distributions. For example, the Singh-Maddala,
Dagum, Fisk (log-logistic), Lomax (Pareto type II),
inverse Lomax, beta distribution of the second kind
distributions are all special cases.
Full details can be found in Kleiber and Kotz (2003), and
Brazauskas (2002). The argument names given here are used
by other families that are special cases of this family.
Fisher scoring is used here and for the special cases too.
The 4-parameter generalized beta II distribution has density
f(y) = a y^{ap-1} / [b^{ap} B(p,q) \{1 + (y/b)^a\}^{p+q}]
for a > 0, b > 0, p > 0, q > 0,
y \geq 0.
Here B is the beta function, and
b is the scale parameter scale,
while the others are shape parameters.
The mean is
E(Y) = b \, \Gamma(p + 1/a) \, \Gamma(q - 1/a) /
(\Gamma(p) \, \Gamma(q))
provided -ap < 1 < aq; these are returned as the fitted values.
Value
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Warning
zero can be a numeric or a character vector
specifying the position(s) or the name(s) (partially or not) of the linear
predictors modeled as intercept–only. Numeric values can be
entered as usual. If names are used, note that the linear predictors in
this family function are
c("scale", "shape1.a", "shape2.p", "shape3.q").
For simplicity, using names rather than numeric vectors is
recommended.
Note
Paramaters "shape1.a", "shape2.p", "shape3.q" are modeled
as intercept only, by default.
If the self-starting initial values fail, try experimenting
with the initial value arguments, iscale, ishape1.a,
ishape2.p and ishape3.q whose default is NULL.
Also, the constraint -ap < 1 < aq may be violated as the
iterations progress so it is worth monitoring convergence, e.g.,
set trace = TRUE.
Successful convergence depends on choosing good initial values. This
process might be difficult for this distribution, since 4 parameters are
involved. Presently, only method = 1 is internally handled to set
initial values. It involves grid search, an internal implementation
of the well-known grid search algorithm for exhaustive searching through
a manually specified subset of the hyperparameter space.
Default value of lss is TRUE standing for the
following order: location (b), shape1.a (a), shape2.p (p),
shape3.q (q). In order to match the arguments of existing R functions,
the option lss = FALSE might be set leading to switch the
position of location (b) and shape1.a (a), only.
Author(s)
T. W. Yee and V. Miranda.
References
Brazauskas, V. (2002)
Fisher information matrix for the Feller-Pareto distribution.
Statistics & Probability Letters,
59, 159–167.
Kleiber, C. and Kotz, S. (2003)
Statistical Size Distributions in Economics and Actuarial Sciences.
Wiley Series in Probability and Statistics.
Hoboken, New Jersey, USA.
McDonald, J. B. and Xu, Y. J. (1995)
A generalization of the beta distribution with applications.
Journal of Econometrics.
66, p.133–152.
McDonald, J. B. (1984)
Some generalized functions for the size distribution of income.
Econometrica,
52, 647–663.
See Also
betaff,
betaII,
dagum,
sinmad,
fisk,
lomax,
inv.lomax,
paralogistic,
inv.paralogistic,
genbetaIIDist.
Examples
#----------------------------------------------------------------------- #
# An example.- In this data set parameters 'shape1.a' and 'shape3.q' are
# generated in terms of x2.
set.seed(1003)
nn <- 200
gdata1 <- data.frame(x2 = runif(nn))
gdata <- transform(gdata1,
y1 = rgen.betaII(nn, scale = exp(1.1), shape1.a = exp(1.2 + x2),
shape2.p = exp(0.7) , shape3.q = exp(2.1 - x2)),
y2 = rgen.betaII(nn, scale = exp(2.0), shape1.a = exp(1.8 + x2),
shape2.p = exp(2.3) , shape3.q = exp(1.9 - x2)),
y3 = rgen.betaII(nn, scale = exp(1.5), shape1.a = exp(1.8),
shape2.p = exp(2.3) , shape3.q = exp(1.3)))
#------------------------------------------------------------------------#
# A single intercept-only model. No covariates.
# Note the use of (optional) initial values.
fit <- vglm(y2 ~ 1, #y3 ~ 1
gen.betaIImr(lss = TRUE,
# OPTIONAL INITIAL VALUES
#iscale = exp(1.5),
#ishape1.a = exp(1.8),
#ishape2.p = exp(2.3),
#ishape3.q = exp(1.3),
imethod = 1),
data = gdata, trace = TRUE, crit = "loglik")
Coef(fit)
coef(fit, matrix = TRUE)
summary(fit)
#------------------------------------------------------------------------#
# An intercept-only model. Two responses.
fit1 <- vglm(cbind(y2, y2) ~ 1, # cbind(y1, y2)
gen.betaIImr(lss = TRUE),
data = gdata, trace = TRUE, crit = "loglik")
Coef(fit1)
coef(fit1, matrix = TRUE)
summary(fit1)
vcov(fit1, untransform = TRUE)
#------------------------------------------------------------------------#
# An example incorporating one covariate. Constraints are set accordingly.
# x2 affects shape1.a and shape3.q.
# Note that the first option uses 'constraints', whilst in the second
# choice we use the argument 'zero' to 'set' the same constraints.
### Option 1.
c1 <- rbind(0, 1, 0, 0)
c2 <- rbind(0, 0, 0, 1)
mycons <- matrix( c(c1, c2), nc = 2, byrow = FALSE)
fit2 <- vglm(y1 ~ x2, gen.betaIImr(lss = TRUE, zero = NULL),
data = gdata, trace = TRUE, crit = "loglik",
constraints = list(x2 = mycons ))
coef(fit2, matrix = TRUE)
summary(fit2)
vcov(fit2)
constraints(fit2)
### Option 2.
fit3 <- vglm(y1 ~ x2,
gen.betaIImr(lss = TRUE,
zero = c("scale", "shape2.p")),
data = gdata, trace = TRUE, crit = "loglik")
coef(fit3, matrix = TRUE)
summary(fit3)
vcov(fit3)
constraints(fit3)
VGAMextra documentation built on Nov. 2, 2023, 5:59 p.m.
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| gen.betaIImr | R Documentation |
Generalized Beta Distribution of the Second Kind family function
Description
Maximum likelihood estimation of the 4-parameter generalized beta II distribution using Fisher scoring.
Usage
gen.betaIImr(lscale = "loglink",
lshape1.a = "loglink",
lshape2.p = "loglink",
lshape3.q = "loglink",
iscale = NULL,
ishape1.a = NULL,
ishape2.p = NULL,
ishape3.q = NULL,
imethod = 1,
lss = TRUE,
gscale = exp(-5:5),
gshape1.a = exp(-5:5),
gshape2.p = exp(-5:5),
gshape3.q = exp(-5:5),
probs.y = c(0.25, 0.50, 0.75),
zero = "shape" )
Arguments
lscale, lshape1.a, lshape2.p, lshape3.q |
Parameter link functions applied to the
shape parameter |
iscale, ishape1.a, ishape2.p, ishape3.q |
Optional initial values for |
imethod |
Initializing method to internally compute the initial values.
Currently, only |
gscale, gshape1.a, gshape2.p, gshape3.q |
Grid search initial values.
See |
zero |
Numeric or Character vector. Position(s) or name(s) of the
parameters/linear predictors to be modeled as intercept-only.
Details at
|
lss, probs.y |
See |
Details
This distribution is most useful for unifying a substantial number of size distributions. For example, the Singh-Maddala, Dagum, Fisk (log-logistic), Lomax (Pareto type II), inverse Lomax, beta distribution of the second kind distributions are all special cases. Full details can be found in Kleiber and Kotz (2003), and Brazauskas (2002). The argument names given here are used by other families that are special cases of this family. Fisher scoring is used here and for the special cases too.
The 4-parameter generalized beta II distribution has density
f(y) = a y^{ap-1} / [b^{ap} B(p,q) \{1 + (y/b)^a\}^{p+q}]
for a > 0, b > 0, p > 0, q > 0,
y \geq 0.
Here B is the beta function, and
b is the scale parameter scale,
while the others are shape parameters.
The mean is
E(Y) = b \, \Gamma(p + 1/a) \, \Gamma(q - 1/a) /
(\Gamma(p) \, \Gamma(q))
provided -ap < 1 < aq; these are returned as the fitted values.
Value
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Warning
zero can be a numeric or a character vector
specifying the position(s) or the name(s) (partially or not) of the linear
predictors modeled as intercept–only. Numeric values can be
entered as usual. If names are used, note that the linear predictors in
this family function are
c("scale", "shape1.a", "shape2.p", "shape3.q").
For simplicity, using names rather than numeric vectors is recommended.
Note
Paramaters "shape1.a", "shape2.p", "shape3.q" are modeled
as intercept only, by default.
If the self-starting initial values fail, try experimenting
with the initial value arguments, iscale, ishape1.a,
ishape2.p and ishape3.q whose default is NULL.
Also, the constraint -ap < 1 < aq may be violated as the
iterations progress so it is worth monitoring convergence, e.g.,
set trace = TRUE.
Successful convergence depends on choosing good initial values. This
process might be difficult for this distribution, since 4 parameters are
involved. Presently, only method = 1 is internally handled to set
initial values. It involves grid search, an internal implementation
of the well-known grid search algorithm for exhaustive searching through
a manually specified subset of the hyperparameter space.
Default value of lss is TRUE standing for the
following order: location (b), shape1.a (a), shape2.p (p),
shape3.q (q). In order to match the arguments of existing R functions,
the option lss = FALSE might be set leading to switch the
position of location (b) and shape1.a (a), only.
Author(s)
T. W. Yee and V. Miranda.
References
Brazauskas, V. (2002) Fisher information matrix for the Feller-Pareto distribution. Statistics & Probability Letters, 59, 159–167.
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Wiley Series in Probability and Statistics. Hoboken, New Jersey, USA.
McDonald, J. B. and Xu, Y. J. (1995) A generalization of the beta distribution with applications. Journal of Econometrics. 66, p.133–152.
McDonald, J. B. (1984) Some generalized functions for the size distribution of income. Econometrica, 52, 647–663.
See Also
betaff,
betaII,
dagum,
sinmad,
fisk,
lomax,
inv.lomax,
paralogistic,
inv.paralogistic,
genbetaIIDist.
Examples
#----------------------------------------------------------------------- #
# An example.- In this data set parameters 'shape1.a' and 'shape3.q' are
# generated in terms of x2.
set.seed(1003)
nn <- 200
gdata1 <- data.frame(x2 = runif(nn))
gdata <- transform(gdata1,
y1 = rgen.betaII(nn, scale = exp(1.1), shape1.a = exp(1.2 + x2),
shape2.p = exp(0.7) , shape3.q = exp(2.1 - x2)),
y2 = rgen.betaII(nn, scale = exp(2.0), shape1.a = exp(1.8 + x2),
shape2.p = exp(2.3) , shape3.q = exp(1.9 - x2)),
y3 = rgen.betaII(nn, scale = exp(1.5), shape1.a = exp(1.8),
shape2.p = exp(2.3) , shape3.q = exp(1.3)))
#------------------------------------------------------------------------#
# A single intercept-only model. No covariates.
# Note the use of (optional) initial values.
fit <- vglm(y2 ~ 1, #y3 ~ 1
gen.betaIImr(lss = TRUE,
# OPTIONAL INITIAL VALUES
#iscale = exp(1.5),
#ishape1.a = exp(1.8),
#ishape2.p = exp(2.3),
#ishape3.q = exp(1.3),
imethod = 1),
data = gdata, trace = TRUE, crit = "loglik")
Coef(fit)
coef(fit, matrix = TRUE)
summary(fit)
#------------------------------------------------------------------------#
# An intercept-only model. Two responses.
fit1 <- vglm(cbind(y2, y2) ~ 1, # cbind(y1, y2)
gen.betaIImr(lss = TRUE),
data = gdata, trace = TRUE, crit = "loglik")
Coef(fit1)
coef(fit1, matrix = TRUE)
summary(fit1)
vcov(fit1, untransform = TRUE)
#------------------------------------------------------------------------#
# An example incorporating one covariate. Constraints are set accordingly.
# x2 affects shape1.a and shape3.q.
# Note that the first option uses 'constraints', whilst in the second
# choice we use the argument 'zero' to 'set' the same constraints.
### Option 1.
c1 <- rbind(0, 1, 0, 0)
c2 <- rbind(0, 0, 0, 1)
mycons <- matrix( c(c1, c2), nc = 2, byrow = FALSE)
fit2 <- vglm(y1 ~ x2, gen.betaIImr(lss = TRUE, zero = NULL),
data = gdata, trace = TRUE, crit = "loglik",
constraints = list(x2 = mycons ))
coef(fit2, matrix = TRUE)
summary(fit2)
vcov(fit2)
constraints(fit2)
### Option 2.
fit3 <- vglm(y1 ~ x2,
gen.betaIImr(lss = TRUE,
zero = c("scale", "shape2.p")),
data = gdata, trace = TRUE, crit = "loglik")
coef(fit3, matrix = TRUE)
summary(fit3)
vcov(fit3)
constraints(fit3)
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