gen.betaIImr: 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" )

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

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.

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.

Author

T. W. Yee and V. Miranda.

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.

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.

Examples

Run this code



#----------------------------------------------------------------------- #
# 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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