Burr {actuar}R Documentation

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 <= x], otherwise, P[X > x].
order order of the moment.
limit limit of the loss variable.

Details

The Burr distribution with parameters shape1 = a, shape2 = b and scale = s has density:

f(x) = (a b (x/s)^b)/(x [1 + (x/s)^b]^(a + 1))

for x > 0, a > 0, b > 0 and s > 0.

The Burr is the distribution of the random variable

s (X/(1 - X))^(1/b),

where X has a Beta distribution with parameters 1 and a.

The Burr distribution has the following special cases:

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

Value

dburr gives the density, pburr gives the distribution function, qburr gives the quantile function, rburr generates random deviates, mburr gives the kth raw moment, and levburr gives the kth moment of the limited loss variable.
Invalid arguments will result in return value NaN, with a warning.

Note

Distribution also known as the Burr Type XII or Singh-Maddala distribution.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

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)
[Package actuar version 1.0-2 Index]