The Single-parameter Pareto Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments, and limited moments for the Single-parameter Pareto distribution with parameter shape.

Usage

dpareto1(x, shape, min, log = FALSE)
ppareto1(q, shape, min, lower.tail = TRUE, log.p = FALSE)
qpareto1(p, shape, min, lower.tail = TRUE, log.p = FALSE)
rpareto1(n, shape, min)
mpareto1(order, shape, min)
levpareto1(limit, shape, min, order = 1)

Arguments

Details

The Single-parameter Pareto distribution with parameter shape = a has density:

f(x) = a b^a/x^(a + 1)

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

Although there appears to be two parameters, only shape is a true parameter. The value of min = b must be set in advance.

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)^k].

Value

dpareto1 gives the density, ppareto1 gives the distribution function, qpareto1 gives the quantile function, rpareto1 generates random deviates, mpareto1 gives the kth raw moment, and levpareto1 gives the kth moment of the limited loss variable.

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

Author(s)

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

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2008), Loss Models, From Data to Decisions, Third Edition, Wiley.

Examples

exp(dpareto1(5, 3, 4, log = TRUE))
p <- (1:10)/10
ppareto1(qpareto1(p, 2, 3), 2, 3)
mpareto1(2, 3, 4) - mpareto(1, 3, 4) ^ 2
levpareto(10, 3, 4, order = 2)

Results