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
$$f(y_1,y_2;a,p,q) = a^{p+q} y_1^{p-1} (y_2-y_1)^{q-1}
\exp(-a y_2) / [\Gamma(p) \Gamma(q)]$$
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.