# 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)
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`.

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.

`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)
``````

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.