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,shape) = exp(-shape * y / mu) y^(shape-1) shape^(shape) /
[mu^(shape) * gamma(shape)]*

for
*mu > 0*,
*shape > 0*
and *y > 0*.
Here,
*gamma()* 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 / shape*. If *0<shape<1* then the density has a
pole at the origin and decreases monotonically as *y* increases.
If *shape=1* then this corresponds to the exponential
distribution. If *shape>1* then the density is zero at the
origin and is unimodal with mode at *y =
mu - mu / shape*; this can be achieved with `lshape="logloglink"`

.

By default, the two linear/additive predictors are
*eta1=log(mu)* and
*eta2=log(shape)*.
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,
`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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