R: Prentice (1974) Log-gamma Distribution

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 = "identitylink", lscale = "loglink",
               lshape = "identitylink", ilocation = NULL, iscale = NULL,
               ishape = NULL, imethod = 1,
               glocation.mux = exp((-4:4)/2), gscale.mux = exp((-4:4)/2),
               gshape = qt(ppoints(6), df = 1), probs.y = 0.3,
               zero = c("scale", "shape"))

### Arguments

`llocation, lscale, lshape` | Parameter link function applied to the location parameter a, positive scale parameter b and the shape parameter q, respectively. See `Links` for more choices.
`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.
`imethod, zero` | See `CommonVGAMffArguments` for information.
`glocation.mux, gscale.mux, gshape, probs.y` | See `CommonVGAMffArguments` for information.

### 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, `digamma`. The mean of Y is a (returned as the fitted values). This is a different parameterization compared to `lgamma3`.

Special cases: q = 0 is the normal distribution with standard deviation b, q = -1 is the extreme value distribution for maximums, 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

`lgamma3`, `lgamma`, `gengamma.stacy`.

### Examples

    pdata <- data.frame(x2 = runif(nn <- 1000))
    pdata <- transform(pdata, loc = -1 + 2*x2, Scale = exp(1))
    pdata <- transform(pdata, y = rlgamma(nn, loc = loc, scale = Scale, shape = 1))
    fit <- vglm(y ~ x2, prentice74(zero = 2:3), data = pdata, trace = TRUE)
    coef(fit, matrix = TRUE)  # Note the coefficients for location

[Package VGAM version 1.1-3 Index]