Renext (version 3.1-4)

Lomax: 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)

Value

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

rlomax generates random deviates.

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 \(\textrm{CV} = \sqrt{2}\).

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")