InverseBurr {actuar}

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)

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.

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

The kth limited moment at some limit d is E[\min(X, d)^k], k > -\tau\gamma and 1 - k/\gamma not a negative integer.

Value

dinvburr gives the density, invburr gives the distribution function, qinvburr gives the quantile function, rinvburr generates random deviates, minvburr gives the kth raw moment, and levinvburr gives the kth moment of the limited loss variable.

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

Note

levinvburr computes the limited expected value using betaint.

Also known as the Dagum distribution. See also Kleiber and Kotz (2003) for alternative names and parametrizations.

The "distributions" package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

Author(s)

Vincent Goulet [e-mail] and Mathieu Pigeon

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)