Lomax {Renext}

Lomax distribution

Description

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

See Also

flomax to fit the Lomax distribution by Maximum Likelihood.

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")