lgammaff: Log-gamma Distribution Family Function
In VGAM: Vector Generalized Linear and Additive Models

lgamma1

R Documentation

Log-gamma Distribution Family Function

Description

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

Usage

lgamma1(lshape = "loglink", ishape = NULL)
lgamma3(llocation = "identitylink", lscale = "loglink",
   lshape = "loglink", ilocation = NULL, iscale = NULL, ishape = 1,
   zero = c("scale", "shape"))

Arguments

`llocation`, `lscale`	Parameter link function applied to the location parameter `a` and the positive scale parameter `b`. See `Links` for more choices.
`lshape`	Parameter link function applied to the positive shape parameter `k`. See `Links` for more choices.
`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. See `CommonVGAMffArguments` for more information.

Details

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

f(y;k)=\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 lgamma1 and the non-standard (3-parameter) log-gamma distribution is fitted with lgamma3.

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.

Examples

ldata <- data.frame(y = rlgamma(100, shape = exp(1)))
fit <- vglm(y ~ 1, lgamma1, ldata, trace = TRUE, crit = "coef")
summary(fit)
coef(fit, matrix = TRUE)
Coef(fit)

ldata <- data.frame(x2 = runif(nn <- 5000))  # Another example
ldata <- transform(ldata, loc = -1 + 2 * x2, Scale = exp(1))
ldata <- transform(ldata, y = rlgamma(nn, loc, sc = Scale, sh = exp(0)))
fit2 <- vglm(y ~ x2, lgamma3, data = ldata, trace = TRUE, crit = "c")
coef(fit2, matrix = TRUE)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.