Density of the Extreme-Value Birnbaum-Saunders distribution

Probability density function of the EVBS distribution on the response (positive) scale. The EVBS variate \(T\) satisfies \(Z = (2/\alpha)\,\sinh\{(\log T - \eta)/2\} \sim \mathrm{GEV}(0,1,\gamma)\).

Usage

devbs(t, eta, alpha, gama, log = FALSE)

Arguments

t

Vector of positive quantiles.

eta

Location on the log scale, typically \(x^\top \beta\).

alpha

Positive shape (scale) parameter.

gama

Extreme-value shape parameter.

log

Logical; if TRUE, the log-density is returned.

Value

A numeric vector of (log-)density values.

Details

The response \(T\) lies in the same max-domain of attraction as the GEV variate. Its extreme-value index is \(2\gamma\) when \(\gamma > 0\), because \(T \approx \beta\alpha^2 Z^2\) in the upper tail, and \(\gamma\) itself when \(\gamma \le 0\): near a finite endpoint the transformation is locally linear (Ferreira, Gomes and Leiva, 2012, Theorems 4.1 and 4.2).

Examples

devbs(c(10, 20, 30), eta = 2.58, alpha = 0.19, gama = -0.16)
#> [1] 0.0385729564 0.0256715224 0.0002807054