FellerPareto: 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)

Value

dfpareto gives the density,

pfpareto gives the distribution function,

qfpareto gives the quantile function,

rfpareto generates random deviates,

mfpareto gives the $k$th raw moment, and

levfpareto gives the $k$th moment of the limited loss variable.

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

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 $P[X \le x]$, otherwise, $P[X > x]$.
order: order of the moment.
limit: limit of the loss variable.

Author

Vincent Goulet vincent.goulet@act.ulaval.ca and Nicholas Langevin

Details

The Feller-Pareto distribution with parameters min $= \mu$, shape1 $= \alpha$, shape2 $= \gamma$, shape3 $= \tau$ and scale $= \theta$, has density: $$f(x) = \frac{\Gamma(\alpha + \tau)}{\Gamma(\alpha)\Gamma(\tau)} \frac{\gamma ((x - \mu)/\theta)^{\gamma \tau - 1}}{% \theta [1 + ((x - \mu)/\theta)^\gamma]^{\alpha + \tau}}$$ for $x > \mu$, $-\infty < \mu < \infty$, $\alpha > 0$, $\gamma > 0$, $\tau > 0$ and $\theta > 0$. (Here $\Gamma(\alpha)$ is the function implemented by R's gamma() and defined in its help.)

The Feller-Pareto is the distribution of the random variable $$\mu + \theta \left(\frac{1 - X}{X}\right)^{1/\gamma},$$ where $X$ has a beta distribution with parameters $\alpha$ and $\tau$.

The Feller-Pareto defines a large family of distributions encompassing the transformed beta family and many variants of the Pareto distribution. Setting $\mu = 0$ 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 $k$th raw moment of the random variable $X$ is $E[X^k]$ for nonnegative integer values of $k < \alpha\gamma$.

The $k$th limited moment at some limit $d$ is $E[\min(X, d)^k]$ for nonnegative integer values of $k$ and $\alpha - j/\gamma$, $j = 1, \dots, k$ not a negative integer.

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

References

Dutang, C., Goulet, V., Langevin, N. (2022). Feller-Pareto and Related Distributions: Numerical Implementation and Actuarial Applications. Journal of Statistical Software, 103(6), 1--22. tools:::Rd_expr_doi("10.18637/jss.v103.i06").

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

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.

Examples

Run this code

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)

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