mckaygamma2: McKay's Bivariate Gamma Distribution

Description

Estimate the two parameters of McKay's bivariate gamma distribution by maximum likelihood estimation.

Usage

mckaygamma2(la = "loge", lp = "loge", lq = "loge",
            ia = NULL, ip = 1, iq = 1, zero = NULL)

Arguments

la, lp, lq

Link functions applied to the (positive) parameters $a$, $p$ and $q$. See Links for more choices.

ia, ip, iq

Initial values for $a$, $p$ and $q$. The default for $a$ is to estimate it using ip and iq.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,3}. The default is none of them.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Details

The joint probability density function is given by

f (y_{1}, y_{2}; a, p, q) = a^{p + q} y_{1}^{p - 1} (y_{2} - y_{1})^{q - 1} \exp (- a y_{2}) / [Γ (p) Γ (q)]

for $a > 0$, $p > 0$, $q > 0$ and $0 < y_1 < y_2$. Here, $\Gamma$ is the gamma function, as in gamma. By default, the linear/additive predictors are $\eta_1=\log(a)$, $\eta_2=\log(p)$, $\eta_3=\log(q)$.

Although Fisher scoring and Newton-Raphson coincide for this distribution, faster convergence may be obtained by choosing better values for the arguments ip and iq.

References

McKay, A. T. (1934) Sampling from batches. Journal of the Royal Statistical Society---Supplement, 1, 207--216.

Kotz, S. and Balakrishnan, N. and Johnson, N. L. (2000) Continuous Multivariate Distributions Volume 1: Models and Applications, 2nd edition, New York: Wiley.

Examples

Run this code

y1 = rgamma(n <- 200, shape=4)
y2 = rgamma(n, shape=8)
ymat = cbind(y1,y2)
fit = vglm(ymat ~ 1, fam=mckaygamma2, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
vcov(fit)
head(fitted(fit))
summary(fit)

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