R: The Transformed Beta Distribution

TransformedBeta {actuar} | R Documentation

## The Transformed Beta Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Transformed Beta distribution with parameters `shape1`, `shape2`, `shape3` and `scale`.

### Usage

    dtrbeta(x, shape1, shape2, shape3, rate = 1, scale = 1/rate,
            log = FALSE)
    ptrbeta(q, shape1, shape2, shape3, rate = 1, scale = 1/rate,
            lower.tail = TRUE, log.p = FALSE)
    qtrbeta(p, shape1, shape2, shape3, rate = 1, scale = 1/rate,
            lower.tail = TRUE, log.p = FALSE)
    rtrbeta(n, shape1, shape2, shape3, rate = 1, scale = 1/rate)
    mtrbeta(order, shape1, shape2, shape3, rate = 1, scale = 1/rate)
    levtrbeta(limit, shape1, shape2, shape3, 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, shape3, 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 beta distribution with parameters `shape1` = a, `shape2` = b, `shape3` = c and `scale` = s, has density:

f(x) = Gamma(a + c)/(Gamma(a) * Gamma(c)) (b (x/s)^(bc))/ (x [1 + (x/s)^b]^(a + c))

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

The transformed beta is the distribution of the random variable

s (X/(1 - X))^(1/b),

where X has a beta distribution with parameters c and a.

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

- A Burr distribution when `shape3 == 1`;

- A loglogistic distribution when `shape1 == shape3 == 1`;

- A paralogistic distribution when `shape3 == 1` and `shape2 == shape1`;

- A generalized Pareto distribution when `shape2 == 1`;

- A Pareto distribution when `shape2 == shape3 == 1`;

- An inverse Burr distribution when `shape1 == 1`;

- An inverse Pareto distribution when `shape2 == shape1 == 1`;

- An inverse paralogistic distribution when `shape1 == 1` and `shape3 == shape2`.

The kth raw moment of the random variable X is E[X^k], -shape3 * shape2 < k < shape1 * shape2.

The kth limited moment at some limit d is E[min(X, d)^k], k > -shape3 * shape2 and shape1 - k/shape2 not a negative integer.

### Value

`dtrbeta` gives the density, `ptrbeta` gives the distribution function, `qtrbeta` gives the quantile function, `rtrbeta` generates random deviates, `mtrbeta` gives the kth raw moment, and `levtrbeta` gives the kth moment of the limited loss variable.

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

### Note

`levtrbeta` computes the limited expected value using `betaint`.

Distribution also known as the Generalized Beta of the Second Kind and Pearson Type VI. 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.

### See Also

`dfpareto` for an equivalent distribution with a location parameter.

### Examples

    exp(dtrbeta(2, 2, 3, 4, 5, log = TRUE))
    p <- (1:10)/10
    ptrbeta(qtrbeta(p, 2, 3, 4, 5), 2, 3, 4, 5)
    qpearson6(0.3, 2, 3, 4, 5, lower.tail = FALSE)

    ## variance
    mtrbeta(2, 2, 3, 4, 5) - mtrbeta(1, 2, 3, 4, 5)^2

    ## case with shape1 - order/shape2 > 0
    levtrbeta(10, 2, 3, 4, scale = 1, order = 2)

    ## case with shape1 - order/shape2 < 0
    levtrbeta(10, 1/3, 0.75, 4, scale = 0.5, order = 2)

[Package actuar version 3.0-0 Index]