mckaygamma2 {VGAM} | R Documentation |
Estimate the two parameters of McKay's bivariate gamma distribution by maximum likelihood estimation.
mckaygamma2(la = "loge", lp = "loge", lq = "loge", ia = NULL, ip = 1, iq = 1, zero = NULL)
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.
|
The joint probability density function is given by
f(y1,y2) = a^(p+q) y1^(p-1) (y2-y1)^(q-1) exp(-a y2) / [gamma(p) gamma(q)]
for a > 0, p > 0, q > 0 and
0<y1<y2.
Here, gamma is the gamma
function, as in gamma
.
By default, the linear/additive predictors are
eta1=log(a),
eta2=log(p),
eta3=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
.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
and vgam
.
The response must be a two column matrix.
Currently, the fitted value is a matrix with two columns; the
first column has values (p+q)/a for the mean of
pmin(y1,y2)
, while the second column is filled with NA
for the unknown mean of pmax(y1,y2)
.
The data are sorted internally and the user need not input the
data presorted.
T. W. Yee
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.
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) fitted(fit)[1:5,] summary(fit)