Lomax distribution

Density function, distribution function, quantile function and random generation for the Lomax distribution.

Usage

dlomax(x, scale = 1.0, shape = 4.0, log = FALSE)
   plomax(q, scale = 1.0, shape = 4.0, lower.tail = TRUE)
   qlomax(p, scale = 1.0, shape = 4.0)
   rlomax(n, scale = 1.0, shape = 4.0)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

scale, shape

Scale and shape parameters. Vectors of length > 1 are not accepted.

log

Logical; if TRUE, the log density is returned.

lower.tail

Logical; if TRUE (default), probabilities are $\textrm{Pr}[X <= x]$, otherwise, $\textrm{Pr}[X > x]$.

Details

The Lomax distribution function with shape $\alpha > 0$ and scale $\beta > 0$ has survival function $$S(y) = \left[1 + y/\beta \right]^{-\alpha} \qquad (y > 0)$$ This distribution has increasing hazard and decreasing mean residual life (MRL). The coefficient of variation decreases with $\alpha$, and tends to $1$ for large $\alpha$. The default value $\alpha=4$ corresponds to $\textrm{CV} = \sqrt{2}$.

Value

dlomax gives the density function, plomax gives the distribution function, qlomax gives the quantile function, and

rlomax generates random deviates.

Note

This distribution is sometimes called log-exponential. It is a special case of Generalised Pareto Distribution (GPD) with positive shape $\xi > 0$, scale $\sigma$ and location $\mu=0$. The Lomax and GPD parameters are related according to $$\alpha = 1/\xi, \qquad \beta = \sigma/\xi.$$ The Lomax distribution can be used in POT to describe excesses following GPD with shape $\xi>0$ thus with decreasing hazard and increasing Mean Residual Life.

Note that the exponential distribution with rate $\nu$ is the limit of a Lomax distribution having large scale $\beta$ and large shape $\alpha$, with the constraint on the shape/scale ratio $\alpha/\beta = \nu$.

References

Johnson N. Kotz S. and N. Balakrishnan Continuous Univariate Distributions vol. 1, Wiley 1994.

Lomax distribution in Wikipedia

Examples

shape <- 5; scale <- 10
xl <- qlomax(c(0.00, 0.99), scale = scale, shape = shape)
x <- seq(from = xl[1], to = xl[2], length.out = 200)
f <- dlomax(x, scale = scale, shape = shape)
plot(x, f, type = "l", main = "Lomax density")

F <- plomax(x, scale = scale, shape = shape)
plot(x, F, type ="l", main ="Lomax distribution function")