mgpd: The MGPD distribution

Description

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

Usage

dmgpd(x, xi, sigma, u, mu, eta, w, log = FALSE)
pmgpd(q, xi, sigma, u, mu, eta, w, lower.tail = TRUE)
qmgpd(p, xi, sigma, u, mu, eta, w, lower.tail = TRUE)
rmgpd(N, xi, sigma, u, mu, eta, w)

Value

dmgpd gives the density, pmgpd gives the distribution function, qmgpd gives the quantile function, and rmgpd generates random deviates. The length of the result is determined by N for rmgpd and by the length of x, q or p otherwise.

Arguments

x, q: vector of quantiles.
xi: shape parameter of the tail GPD (scalar).
sigma: scale parameter of the tail GPD (scalar).
u: threshold parameter of the tail GPD (scalar).
mu: means of the gamma mixture components (vector).
eta: shapes of the gamma mixture components (vector).
w: weights of the gamma mixture components (vector). Must sum to one.
log: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are $P(X\leq x)$ otherwise $P(X>x)$.
p: vector of probabilities.
N: number of observations.

Details

The MGPD distribution is an extreme value mixture model with density $$f_{MGPD}(x|\xi,\sigma,u,\mu,\eta,w)=\left\{\begin{array}{ll} f_{MG}(x|\mu,\eta,w), & x\leq u \\ (1-F_{MG}(u|\mu,\eta,w))f_{GPD}(x|\xi,\sigma,u), &\mbox{otherwise}, \end{array}\right.$$ where $f_{MG}$ is the density of the mixture of Gammas, $F_{MG}$ is the distribution function of the mixture of Gammas and $f_{GPD}$ is the density of the Generalized Pareto Distribution, i.e. $$f_{GPD}(x|\xi,\sigma,u)=\left\{\begin{array}{ll} 1- (1+\frac{\xi}{\sigma}(x-u))^{-1/\xi}, & \mbox{if } \xi\neq 0,\\ 1- \exp\left(-\frac{x-u}{\sigma}\right), & \mbox{if } \xi = 0, \end{array}\right.$$ where $\xi$ is a shape parameter, $\sigma > 0$ is a scale parameter and $u>0$ is a threshold.

References

do Nascimento, Fernando Ferraz, Dani Gamerman, and Hedibert Freitas Lopes. "A semiparametric Bayesian approach to extreme value estimation." Statistics and Computing 22.2 (2012): 661-675.

Examples

Run this code

dmgpd(3, xi = 0.5, sigma = 2,5, u = 5, mu = c(2,3), eta = c(1,2), w = c(0.3,0.7))

Run the code above in your browser using DataLab