actuar (version 0.9-3)

Burr: The Burr Distribution

Description

Density, 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)

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.

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.

Details

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

• ALoglogisticdistribution whenshape1 == 1;
• AParalogisticdistribution whenshape2 == shape1;
• AParetodistribution whenshape2 == 1.

The $k$th raw moment of the random variable $X$ is $E[X^k]$ and the $k$ limited moment at some limit $d$ is $E[\min(X, d)]$.

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second Edition, Wiley.

Examples

exp(dburr(2, 3, 4, 5, log = TRUE))
p <- (1:10)/10
pburr(qburr(p, 2, 3, 1), 2, 3, 1)
mburr(2, 1, 2, 3) - mburr(1, 1, 2, 3) ^ 2
levburr(10, 1, 2, 3, order = 2)