actuar (version 0.9-3)

InverseBurr: The Inverse Burr Distribution

Description

Density, 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.

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.

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:

• ALoglogisticdistribution whenshape1 == 1;
• AnInverse Paretodistribution whenshape2 == 1;
• AnInverse Paralogisticdistribution whenshape1 == shape2.

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(dinvburr(2, 3, 4, 5, log = TRUE))
p <- (1:10)/10
pinvburr(qinvburr(p, 2, 3, 1), 2, 3, 1)
minvburr(2, 1, 2, 3) - minvburr(1, 1, 2, 3) ^ 2
levinvburr(10, 1, 2, 3, order = 2)