VGAM (version 1.1-13)

lgamma1: 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"))

Value

An object of class "vglmff"

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

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.

Author

T. W. Yee

Details

The probability density function of the standard log-gamma distribution is given by $$f(y;k)=\exp [ky-\exp (y)]/\mathrm{\Gamma }(k),$$ 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 $$f(y)=\exp [k(y-a)/b-\exp ((y-a)/b)]/(b\mathrm{\Gamma }(k)).$$ The mean and variance of $Y$ are a + b*digamma(k) (returned as the fitted values) and b^2 * trigamma(k), respectively.

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, gengamma.stacy, prentice74, gamma1, lgamma.

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)