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 0.7-9 Index]