Burr: The Burr Distribution

Description

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

Usage

dburr(x, shape1, shape2, rate = 1, scale = 1/rate,
      log = FALSE)
pburr(q, shape1, shape2, rate = 1, scale = 1/rate,
      lower.tail = TRUE, log.p = FALSE)
qburr(p, shape1, shape2, rate = 1, scale = 1/rate,
      lower.tail = TRUE, log.p = FALSE)
rburr(n, shape1, shape2, rate = 1, scale = 1/rate)
mburr(order, shape1, shape2, rate = 1, scale = 1/rate)
levburr(limit, shape1, shape2, rate = 1, scale = 1/rate,
        order = 1)

Value

dburr gives the density,

pburr gives the distribution function,

qburr gives the quantile function,

rburr generates random deviates,

mburr gives the $k$th raw moment, and

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

Author

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

Details

The Burr distribution with parameters shape1 $= \alpha$, shape2 $= \gamma$ and scale $= \theta$ has density: $$f(x) = \frac{\alpha \gamma (x/\theta)^\gamma}{% x [1 + (x/\theta)^\gamma]^{\alpha + 1}}$$ for $x > 0$, $\alpha > 0$, $\gamma > 0$ and $\theta > 0$.

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

The Burr distribution has the following special cases:

A Loglogistic distribution when shape1 == 1;
A Paralogistic distribution when shape2 == shape1;
A Pareto distribution when shape2 == 1.

The $k$th raw moment of the random variable $X$ is $E[X^k]$, $-\gamma < k < \alpha\gamma$.

The $k$th limited moment at some limit $d$ is $E[\min(X, d)^k]$, $k > -\gamma$ and $\alpha - k/\gamma$ not a negative integer.

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

Run this code

exp(dburr(1, 2, 3, log = TRUE))
p <- (1:10)/10
pburr(qburr(p, 2, 3, 2), 2, 3, 2)

## variance
mburr(2, 2, 3, 1) - mburr(1, 2, 3, 1) ^ 2

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

## case with shape1 - order/shape2 < 0
levburr(10, 1.5, 0.5, 1, order = 2)

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