actuar (version 3.3-4)

InverseBurr: The Inverse Burr Distribution

Description

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

Usage

dinvburr(x, shape1, shape2, rate = 1, scale = 1/rate,
         log = FALSE)
pinvburr(q, shape1, shape2, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
qinvburr(p, shape1, shape2, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
rinvburr(n, shape1, shape2, rate = 1, scale = 1/rate)
minvburr(order, shape1, shape2, rate = 1, scale = 1/rate)
levinvburr(limit, shape1, shape2, rate = 1, scale = 1/rate,
           order = 1)

Value

dinvburr gives the density,

invburr gives the distribution function,

qinvburr gives the quantile function,

rinvburr generates random deviates,

minvburr gives the \(k\)th raw moment, and

levinvburr 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 and Mathieu Pigeon

Details

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

The inverse 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 \(\tau\) and \(1\).

The inverse Burr distribution has the following special cases:

• A Loglogistic distribution when shape1 == 1;

• An Inverse Pareto distribution when shape2 == 1;

• An Inverse Paralogistic distribution when shape1 == shape2.

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

The \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)^k]\), \(k > -\tau\gamma\) and \(1 - k/\gamma\) 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

exp(dinvburr(2, 2, 3, 1, log = TRUE))
p <- (1:10)/10
pinvburr(qinvburr(p, 2, 3, 1), 2, 3, 1)

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

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

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