actuar (version 3.0-0)

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)

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: $$f(x) = \frac{\alpha \gamma (x/\theta)^\gamma}{% \theta [1 + (x/\theta)^\gamma]^{\alpha + 1}}$$ 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.

See Also

dpareto4 for an equivalent distribution with a location parameter.

Examples

# NOT RUN {
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)
# }