VGAM (version 0.9-2)

gamma2.ab: 2-parameter Gamma Distribution

Description

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

Usage

gamma2.ab(lrate = "loge", lshape = "loge",
          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.

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

Value

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

Details

The density function is given by $$f(y) = \exp(-rate \times y) \times y^{shape-1} \times rate^{shape} / \Gamma(shape)$$ 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 $\eta_1 = \log(rate)$ and $\eta_2 = \log(shape)$.

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 The observed information matrix corresponds to the Newton-Raphson algorithm and may be withdrawn from the family function in the future.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

References

Most standard texts on statistical distributions describe the 2-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.

See Also

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

Examples

# Essentially a 1-parameter gamma
gdata <- data.frame(y1 = rgamma(n <- 100, shape =  exp(1)))
fit1 <- vglm(y1 ~ 1, gamma1, gdata, trace = TRUE)
fit2 <- vglm(y1 ~ 1, gamma2.ab, gdata, trace = TRUE, crit = "coef")
coef(fit2, matrix = TRUE)
Coef(fit2)

# Essentially a 2-parameter gamma
gdata <- data.frame(y2 = rgamma(n = 500, rate = exp(1), shape = exp(2)))
fit2 <- vglm(y2 ~ 1, gamma2.ab, gdata, trace = TRUE, crit = "coef")
coef(fit2, matrix = TRUE)
Coef(fit2)
summary(fit2)