R: The Feller Pareto Distribution

FellerPareto {actuar} | R Documentation

## The Feller Pareto Distribution

### Description

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

### Usage

    dfpareto(x, min, shape1, shape2, shape3, rate = 1, scale = 1/rate,
            log = FALSE)
    pfpareto(q, min, shape1, shape2, shape3, rate = 1, scale = 1/rate,
            lower.tail = TRUE, log.p = FALSE)
    qfpareto(p, min, shape1, shape2, shape3, rate = 1, scale = 1/rate,
            lower.tail = TRUE, log.p = FALSE)
    rfpareto(n, min, shape1, shape2, shape3, rate = 1, scale = 1/rate)
    mfpareto(order, min, shape1, shape2, shape3, rate = 1, scale = 1/rate)
    levfpareto(limit, min, 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.
`min` | lower bound of the support of the distribution.
`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 Feller-Pareto distribution with parameters `min` = m, `shape1` = a, `shape2` = b, `shape3` = c and `scale` = s, has density:

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

for x > m, -Inf < m < Inf, 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 Feller-Pareto is the distribution of the random variable

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

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

The Feller-Pareto defines a large family of distributions encompassing the transformed beta family and many variants of the Pareto distribution. Setting min = 0 yields the transformed beta distribution.

The Feller-Pareto distribution also has the following direct special cases:

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

- A Pareto III distribution when `shape1 shape3 == 1`;

- A Pareto II distribution when `shape1 shape2 == 1`;

- A Pareto I distribution when `shape1 shape2 == 1` and `min = scale`.

The kth raw moment of the random variable X is E[X^k] for nonnegative integer values of k < shape1 * shape2.

The kth limited moment at some limit d is E[min(X, d)^k] for nonnegative integer values of k and shape1 - j/shape2, j = 1, …, k not a negative integer.

Note that the range of admissible values for k in raw and limited moments is larger when min == 0.

### Value

`dfpareto` gives the density, `pfpareto` gives the distribution function, `qfpareto` gives the quantile function, `rfpareto` generates random deviates, `mfpareto` gives the kth raw moment, and `levfpareto` gives the kth moment of the limited loss variable.

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

### Note

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

For the Feller-Pareto and other Pareto distributions, we use the classification of Arnold (2015) with the parametrization of Klugman et al. (2012).

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 Nicholas Langevin

### References

Arnold, B.C. (2015), Pareto Distributions, Second Edition, CRC Press.

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.

Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, Dover.

### See Also

`dtrbeta` for the transformed beta distribution.

### Examples

    exp(dfpareto(2, 1, 2, 3, 4, 5, log = TRUE))
    p <- (1:10)/10
    pfpareto(qfpareto(p, 1, 2, 3, 4, 5), 1, 2, 3, 4, 5)

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

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

    ## case with shape1 - order/shape2 < 0
    levfpareto(20, 10, 0.1, 14, 2, scale = 1.5, order = 2)

[Package actuar version 3.1-2 Index]