VGAM (version 0.7-5)

prentice74: 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.

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.

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 $q \ne 0$, 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

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