mgamma: The Gamma Mixture Distribution

In manueleleonelli/extrememix: Bayesian Estimation of Extreme Value Mixture Models

The Gamma Mixture Distribution

Description

Density, distribution function, quantile function and random generation for the mixture of Gamma distribution.

Usage

dmgamma(x, mu, eta, w, log = FALSE)

pmgamma(q, mu, eta, w, lower.tail = TRUE)

qmgamma(p, mu, eta, w, lower.tail = TRUE)

rmgamma(N, mu, eta, w)

Arguments

x, q	vector of quantiles.
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 Gamma distribution has density

f_{GA}(x|\mu,\eta)= \frac{(\eta/\mu)^\eta}{\Gamma(\eta)}x^{\eta-1}\exp(-(\eta/\mu)x), \hspace{1cm} x>0,

where \mu>0 is the mean of the distribution and \eta>0 is its shape. The density of a mixture of Gamma distributions with k components is defined as

f_{MG}(x|\mu,\eta,w)=\sum_{i=1}^k w_if_{GA}(x|\mu_i,\eta_i),

where w_i,\mu_i,\eta_i >0, for i=1,\dots,k, w_1+\cdots+w_k=1, \mu=(\mu_1,\dots,\mu_k), \eta = (\eta_1,\dots,\eta_k) and w=(w_1,\dots,w_k).

Value

dmgamma gives the density, pmgamma gives the distribution function, qmgamma gives the quantile function, and rmgamma generates random deviates.

The length of the result is determined by N for rmgamma and by the length of x, q or p otherwise.

References

Wiper, Michael, David Rios Insua, and Fabrizio Ruggeri. "Mixtures of gamma distributions with applications." Journal of Computational and Graphical Statistics 10.3 (2001): 440-454.

Examples

dmgamma(3, mu = c(2,3), eta = c(1,2), w = c(0.3,0.7))

manueleleonelli/extrememix documentation built on Oct. 25, 2024, 6:24 p.m.