# prentice74: Prentice (1974) Log-gamma Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3 4 5 6``` ```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.

T. W. Yee

## References

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

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

## Examples

 ```1 2 3 4 5``` ```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 ```

### Example output

```Loading required package: stats4