View source: R/family.univariate.R

gamma2 | R Documentation |

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

```
gamma2(lmu = "loglink", lshape = "loglink", imethod = 1, ishape = NULL,
parallel = FALSE, deviance.arg = FALSE, zero = "shape")
```

`lmu, lshape` |
Link functions applied to the (positive) |

`ishape` |
Optional initial value for |

`imethod` |
An integer with value |

`deviance.arg` |
Logical. If |

`zero` |
See |

`parallel` |
Details at |

This distribution can model continuous skewed responses. The density function is given by

```
f(y;\mu,a) = \frac{\exp(-a y / \mu) \times
(a y / \mu)^{a-1}
\times a}{
\mu \times \Gamma(a)}
```

for
`\mu > 0`

,
`a > 0`

and `y > 0`

.
Here,
`\Gamma(\cdot)`

is the gamma
function, as in `gamma`

.
The mean of *Y* is `\mu=\mu`

(returned as the fitted
values) with variance `\sigma^2 = \mu^2 / a`

. If `0<a<1`

then the density has a
pole at the origin and decreases monotonically as `y`

increases.
If `a=1`

then this corresponds to the exponential
distribution. If `a>1`

then the density is zero at the
origin and is unimodal with mode at `y = \mu - \mu / a`

; this can be achieved with `lshape="logloglink"`

.

By default, the two linear/additive predictors are
`\eta_1=\log(\mu)`

and
`\eta_2=\log(a)`

.
This family function implements Fisher scoring and the working
weight matrices are diagonal.

This VGAM family function handles *multivariate* responses,
so that a matrix can be used as the response. The number of columns is
the number of species, say, and `zero=-2`

means that *all*
species have a shape parameter equalling a (different) intercept only.

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

and `vgam`

.

The response must be strictly positive. A moment estimator for the shape parameter may be implemented in the future.

If `mu`

and `shape`

are vectors, then ```
rgamma(n = n,
shape = shape, scale = mu/shape)
```

will generate random gamma variates of this
parameterization, etc.;
see `GammaDist`

.

T. W. Yee

The parameterization of this VGAM family function is the 2-parameter gamma distribution described in the monograph

McCullagh, P. and Nelder, J. A. (1989).
*Generalized Linear Models*, 2nd ed. London: Chapman & Hall.

`gamma1`

for the 1-parameter gamma distribution,
`gammaR`

for another parameterization of
the 2-parameter gamma distribution that is directly matched
with `rgamma`

,
`bigamma.mckay`

for *a* bivariate gamma distribution,
`gammaff.mm`

for another,
`expexpff`

,
`GammaDist`

,
`gordlink`

,
`CommonVGAMffArguments`

,
`simulate.vlm`

,
`negloglink`

.

```
# Essentially a 1-parameter gamma
gdata <- data.frame(y = rgamma(n = 100, shape = exp(1)))
fit1 <- vglm(y ~ 1, gamma1, data = gdata)
fit2 <- vglm(y ~ 1, gamma2, data = gdata, trace = TRUE, crit = "coef")
coef(fit2, matrix = TRUE)
c(Coef(fit2), colMeans(gdata))
# Essentially a 2-parameter gamma
gdata <- data.frame(y = rgamma(n = 500, rate = exp(-1), shape = exp(2)))
fit2 <- vglm(y ~ 1, gamma2, data = gdata, trace = TRUE, crit = "coef")
coef(fit2, matrix = TRUE)
c(Coef(fit2), colMeans(gdata))
summary(fit2)
```

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