Renext (version 3.1-4)

Maxlo: 'maxlo' distribution

Description

Density function, distribution function, quantile function and random generation for the 'maxlo' distribution.

Usage

dmaxlo(x, scale = 1.0, shape = 4.0, log = FALSE)
   pmaxlo(q, scale = 1.0, shape = 4.0, lower.tail = TRUE)
   qmaxlo(p, scale = 1.0, shape = 4.0)
   rmaxlo(n, scale = 1.0, shape = 4.0)

Value

dmaxlo gives the density function, pmaxlo gives the distribution function, qmaxlo gives the quantile function, and

rmaxlo generates random deviates.

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

scale, shape

Shift 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 'maxlo' distribution function with shape \(\alpha>0\) and scale \(\beta>0\) is a special case of Generalised Pareto (GPD) with negative shape \(\xi < 0\) and location at zero. This is the finite upper endpoint case of the GPD. Its name is nonstandard and was chosen to suggest some form of symmetry with respect to the Lomax distribution.

The survival function is $$S(y) = \left[1-y/\beta\right]^\alpha \qquad 0 < y < \beta$$

This distribution has a coefficient of variation smaller than \(1\).

See Also

fmaxlo to fit such a distribution by Maximum Likelihood.

Examples

xs <- rmaxlo(500, shape = 2.2, scale = 1000)
hist(xs, main = "'maxlo' distribution"); rug(xs)

xs <- rmaxlo(500, shape = 4, scale = 1000)
hist(xs, main = "'maxlo' distribution"); rug(xs)

x <- seq(from = -10, to = 1010, by = 2)
plot(x = x, y = dmaxlo(x, shape = 4, scale = 1000),
     type = "l", ylab = "dens",
     col = "orangered", main = "dmaxlo and dgpd")
abline(h = 0)
lines(x = x, y = dgpd(x, shape = -1/4, scale = 250),
     type = "l",
     col = "SpringGreen3", lty = "dashed")