prentice74 {VGAM}R Documentation

Prentice (1974) Log-gamma Distribution

Description

Estimation of a 3-parameter log-gamma distribution described by Prentice (1974).

Usage

prentice74(llocation="identity", lscale="loge", lshape="identity",
           elocation=list(), escale=list(), eshape=list(),
           ilocation=NULL, iscale=NULL, ishape=NULL, zero=NULL)

Arguments

llocation Parameter link function applied to the location parameter a. See Links for more choices.
lscale Parameter link function applied to the positive scale parameter b. See Links for more choices.
lshape Parameter link function applied to the shape parameter q. See Links for more choices.
elocation, escale, eshape List. Extra argument for each of the links. See earg in Links for general information.
ilocation, iscale Initial value for a and b, respectively. The defaults mean an initial value is determined internally for each.
ishape Initial value for q. If failure to converge occurs, try some other value. The default means an initial value is determined internally.
zero An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,3}. The default value means none are modelled as intercept-only terms.

Details

The probability density function is given by

f(y;a,b,q) = |q| * exp(w/q^2 - e^w) / (b*gamma(1/q^2)),

for shape parameter q != 0, positive scale parameter b > 0, location parameter a, and all real y. Here, w = (y-a)*q/b+psi(1/q^2) where psi is the digamma function. The mean of Y is a (returned as the fitted values). This is a different parameterization compared to lgamma3ff.

Special cases: q=0 is the normal distribution with standard deviation b, q=-1 is the extreme value distribution for maxima, q=1 is the extreme value distribution for minima (Weibull). If q>0 then the distribution is left skew, else q<0 is right skew.

Value

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

Warning

The special case q=0 is not handled, therefore estimates of q too close to zero may cause numerical problems.

Note

The notation used here differs from Prentice (1974): alpha=a, sigma=b. Fisher scoring is used.

Author(s)

T. W. Yee

References

Prentice, R. L. (1974) A log gamma model and its maximum likelihood estimation. Biometrika, 61, 539–544.

See Also

lgamma3ff, lgamma.

Examples

x = runif(n <- 5000)
loc = -1 + 2*x
Scale = exp(1+x)
y = rlgamma(n, loc=loc, scale=Scale, k=1)
fit = vglm(y ~ x, prentice74(zero=3), trace=TRUE)
coef(fit, matrix=TRUE)  # Note the coefficients for location

[Package VGAM version 0.7-7 Index]