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 The observed information matrix corresponds to the Newton-Raphson algorithm and may be withdrawn from the family function in the future.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 0.7-7 Index]