lgammaff {VGAM} | R Documentation

## Log-gamma Distribution Family Function

### Description

Estimation of the parameter of the standard and nonstandard log-gamma distribution.

### Usage

    lgammaff(link = "loge", earg=list(), init.k = NULL)
    lgamma3ff(llocation="identity", lscale="loge", lshape="loge",
              elocation=list(), escale=list(), eshape=list(),
              ilocation=NULL, iscale=NULL, ishape=1, zero=NULL)

### Arguments

`llocation` | Parameter link function applied to the location parameter a. See `Links` for more choices.
`lscale` | Parameter link function applied to the positive scale parameter b. See `Links` for more choices.
`link, lshape` | Parameter link function applied to the positive shape parameter k. See `Links` for more choices.
`earg, elocation, escale, eshape` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`init.k, ishape` | Initial value for k. If given, it must be positive. If failure to converge occurs, try some other value. The default means an initial value is determined internally.
`ilocation, iscale` | Initial value for a and b. The defaults mean an initial value is determined internally for each.
`zero` | An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,3}. The default value means none are modelled as intercept-only terms.

### Details

The probability density function of the standard log-gamma distribution is given by

f(y) = exp[ky - exp(y)]/gamma(k),

for parameter k>0 and all real y. The mean of Y is `digamma(k)` (returned as the fitted values) and its variance is `trigamma(k)`.

For the non-standard log-gamma distribution, one replaces y by (y-a)/b, where a is the location parameter and b is the positive scale parameter. Then the density function is

f(y) = exp[k(y-a)/b - exp((y-a)/b)]/(b*gamma(k)).

The mean and variance of Y are `a + b*digamma(k)` (returned as the fitted values) and `b^2 * trigamma(k)`, respectively.

### Value

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

### Note

The standard log-gamma distribution can be viewed as a generalization of the standard type 1 extreme value density: when k=1 the distribution of -Y is the standard type 1 extreme value distribution.

The standard log-gamma distribution is fitted with `lgammaff` and the non-standard (3-parameter) log-gamma distribution is fitted with `lgamma3ff`.

### Author(s)

T. W. Yee

### References

Kotz, S. and Nadarajah, S. (2000) Extreme Value Distributions: Theory and Applications, pages 48–49, London: Imperial College Press.

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, 2nd edition, Volume 2, p.89, New York: Wiley.

### See Also

`rlgamma`, `ggamma`, `prentice74`, `gamma1`, `lgamma`.

### Examples

    y = rlgamma(n <- 100, k=exp(1))
    fit = vglm(y ~ 1, lgammaff, trace=TRUE, crit="c")
    summary(fit)
    coef(fit, matrix=TRUE)
    Coef(fit)

    # Another example
    x = runif(n <- 5000)
    loc = -1 + 2*x
    Scale = exp(1+x)
    y = rlgamma(n, loc=loc, scale=Scale, k=exp(0))
    fit = vglm(y ~ x, lgamma3ff(zero=3), trace=TRUE, crit="c")
    coef(fit, matrix=TRUE)

[Package VGAM version 0.7-9 Index]