R: The Transformed Gamma Distribution

TransformedGamma {actuar} | R Documentation

## The Transformed Gamma Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Transformed Gamma distribution with parameters `shape1`, `shape2` and `scale`.

### Usage

    dtrgamma(x, shape1, shape2, rate = 1, scale = 1/rate,
             log = FALSE)
    ptrgamma(q, shape1, shape2, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    qtrgamma(p, shape1, shape2, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    rtrgamma(n, shape1, shape2, rate = 1, scale = 1/rate)
    mtrgamma(order, shape1, shape2, rate = 1, scale = 1/rate)
    levtrgamma(limit, shape1, shape2, rate = 1, scale = 1/rate,
               order = 1)

### Arguments

`x, q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1`, the length is taken to be the number required.
`shape1, shape2, scale` | parameters. Must be strictly positive.
`rate` | an alternative way to specify the scale.
`log, log.p` | logical; if `TRUE`, probabilities/densities p are returned as log(p).
`lower.tail` | logical; if `TRUE` (default), probabilities are P[X <= x], otherwise, P[X > x].
`order` | order of the moment.
`limit` | limit of the loss variable.

### Details

The transformed gamma distribution with parameters `shape1` = a, `shape2` = b and `scale` = s has density:

f(x) = b u^a exp(-u) / (x Gamma(a)), u = (x/s)^b

for x > 0, a > 0, b > 0 and s > 0. (Here Gamma(a) is the function implemented by R's `gamma()` and defined in its help.)

The transformed gamma is the distribution of the random variable s X^(1/b), where X has a gamma distribution with shape parameter a and scale parameter 1 or, equivalently, of the random variable Y^(1/b) with Y a gamma distribution with shape parameter a and scale parameter s^b.

The transformed gamma probability distribution defines a family of distributions with the following special cases:

- A Gamma distribution when `shape2 == 1`;

- A Weibull distribution when `shape1 == 1`;

- An Exponential distribution when `shape2 == shape1 == 1`.

The kth raw moment of the random variable X is E[X^k] and the kth limited moment at some limit d is E[min(X, d)^k], k > -shape1 * shape2.

### Value

`dtrgamma` gives the density, `ptrgamma` gives the distribution function, `qtrgamma` gives the quantile function, `rtrgamma` generates random deviates, `mtrgamma` gives the kth raw moment, and `levtrgamma` gives the kth moment of the limited loss variable.

Invalid arguments will result in return value `NaN`, with a warning.

### Note

Distribution also known as the Generalized Gamma. See also Kleiber and Kotz (2003) for alternative names and parametrizations.

The `"distributions"` package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

### Examples

    exp(dtrgamma(2, 3, 4, 5, log = TRUE))
    p <- (1:10)/10
    ptrgamma(qtrgamma(p, 2, 3, 4), 2, 3, 4)
    mtrgamma(2, 3, 4, 5) - mtrgamma(1, 3, 4, 5) ^ 2
    levtrgamma(10, 3, 4, 5, order = 2)

[Package actuar version 3.0-0 Index]