gamma2.ab {VGAM} | R Documentation

## 2-parameter Gamma Distribution

### Description

Estimates the 2-parameter gamma distribution by maximum likelihood estimation.

### Usage

    gamma2.ab(lrate = "loge", lshape = "loge",
              erate=list(), eshape=list(),
              irate=NULL, ishape=NULL, expected = TRUE, zero = 2)

### Arguments

`lrate, lshape` | Link functions applied to the (positive) rate and shape parameters. See `Links` for more choices.
`erate, eshape` | List. Extra arguments for the links. See `earg` in `Links` for general information.
`expected` | Logical. Use Fisher scoring? The default is yes, otherwise Newton-Raphson is used.
`irate, ishape` | Optional initial values for rate and shape. A `NULL` means a value is computed internally. If a failure to converge occurs, try using these arguments.
`zero` | An integer specifying which linear/additive predictor is to be modelled as an intercept only. If assigned, the single value should be either 1 or 2 or `NULL`. The default is to model shape as an intercept only. A value `NULL` means neither 1 or 2.

### Details

The density function is given by

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

for shape > 0, rate > 0 and y > 0. Here, gamma(shape) is the gamma function, as in `gamma`. The mean of Y is mu=shape/rate (returned as the fitted values) with variance sigma^2 = mu^2 /shape = shape/rate^2. By default, the two linear/additive predictors are eta1=log(rate) and eta2=log(shape).

The argument `expected` refers to the type of information matrix. The expected information matrix corresponds to Fisher scoring and is numerically better here. The observed information matrix corresponds to the Newton-Raphson algorithm and may be withdrawn from the family function in the future. If both algorithms work then the differences in the results are often not huge.

### Value

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

### Note

The parameters rate and shape match with the arguments `rate` and `shape` of `rgamma`. Often, scale=1/rate is used.

If rate=1 use the family function `gamma1` to estimate shape.

### Author(s)

T. W. Yee

### References

Most standard texts on statistical distributions describe the 2-parameter gamma distribution, e.g.,

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

### See Also

`gamma1` for the 1-parameter gamma distribution, `gamma2` for another parameterization of the 2-parameter gamma distribution, `mckaygamma2` for a bivariate gamma distribution, `expexp`.

### Examples

    # Essentially a 1-parameter gamma
    y = rgamma(n <- 100, shape= exp(1))
    fit1 = vglm(y ~ 1, gamma1, trace=TRUE, crit="c")
    fit2 = vglm(y ~ 1, gamma2.ab, trace=TRUE, crit="c")
    coef(fit1, matrix=TRUE)
    Coef(fit1)
    coef(fit2, matrix=TRUE)
    Coef(fit2)

    # Essentially a 2-parameter gamma
    y = rgamma(n <- 500, rate=exp(1), shape=exp(2))
    fit2 = vglm(y ~ 1, gamma2.ab, trace=TRUE, crit="c")
    coef(fit2, matrix=TRUE)
    Coef(fit2)
    summary(fit2)

[Package VGAM version 0.7-9 Index]