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

## Description

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

## Usage

 ```1 2 3 4``` ```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`.

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`.
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```ldata <- data.frame(y = rlgamma(100, shape = exp(1))) fit <- vglm(y ~ 1, lgamma1, data = 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, scale = Scale, shape = exp(0))) fit2 <- vglm(y ~ x2, lgamma3, data = ldata, trace = TRUE, crit = "c") coef(fit2, matrix = TRUE) ```