VGAM (version 1.0-4)

# lgamma1: Log-gamma Distribution Family Function

## Description

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

## Usage

```lgamma1(lshape = "loge", ishape = NULL)
lgamma3(llocation = "identitylink", lscale = "loge", lshape = "loge",
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.

## Value

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

## 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.

## 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

Run this code
```# NOT RUN {
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)
# }
```

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