# gamma1: 1-parameter Gamma Regression Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

Estimates the 1-parameter gamma distribution by maximum likelihood estimation.

## Usage

 ```1 2 3``` ```gamma1(link = "loglink", zero = NULL, parallel = FALSE, type.fitted = c("mean", "percentiles", "Qlink"), percentiles = 50) ```

## Arguments

 `link` Link function applied to the (positive) shape parameter. See `Links` for more choices and general information. `zero, parallel` Details at `CommonVGAMffArguments`. `type.fitted, percentiles` See `CommonVGAMffArguments` for information. Using `"Qlink"` is for quantile-links in VGAMextra.

## Details

The density function is given by

f(y) = exp(-y) y^(shape-1) / gamma(shape)

for shape > 0 and y > 0. Here, gamma(shape) is the gamma function, as in `gamma`. The mean of Y (returned as the default fitted values) is mu=shape, and the variance is sigma^2 = shape.

## Value

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

## Note

This VGAM family function can handle a multiple responses, which is inputted as a matrix.

The parameter shape matches with `shape` in `rgamma`. The argument `rate` in `rgamma` is assumed 1 for this family function, so that `scale = 1` is used for calls to `dgamma`, `qgamma`, etc.

If rate is unknown use the family function `gammaR` to estimate it too.

T. W. Yee

## References

Most standard texts on statistical distributions describe the 1-parameter gamma distribution, e.g.,

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

`gammaR` for the 2-parameter gamma distribution, `lgamma1`, `lindley`, `simulate.vlm`.
 ```1 2 3 4 5``` ```gdata <- data.frame(y = rgamma(n = 100, shape = exp(3))) fit <- vglm(y ~ 1, gamma1, data = gdata, trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) summary(fit) ```