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 0.7-7 Index]