bigamma.mckay | R Documentation |
Estimate the three parameters of McKay's bivariate gamma distribution by maximum likelihood estimation.
bigamma.mckay(lscale = "loglink", lshape1 = "loglink",
lshape2 = "loglink", iscale = NULL, ishape1 = NULL,
ishape2 = NULL, imethod = 1, zero = "shape")
lscale , lshape1 , lshape2 |
Link functions applied to the (positive)
parameters |
iscale , ishape1 , ishape2 |
Optional initial values for |
imethod , zero |
See |
One of the earliest forms of the bivariate gamma distribution has a joint probability density function given by
f(y_1,y_2;a,p,q) =
(1/a)^{p+q} y_1^{p-1} (y_2-y_1)^{q-1}
\exp(-y_2 / a) / [\Gamma(p) \Gamma(q)]
for a > 0
, p > 0
, q > 0
and
0 < y_1 < y_2
(Mckay, 1934).
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)
.
The marginal distributions are gamma,
with shape parameters p
and p+q
respectively, but they have a
common scale parameter a
.
Pearson's product-moment correlation coefficient
of y_1
and y_2
is
\sqrt{p/(p+q)}
.
This distribution is also
known as the bivariate Pearson type III distribution.
Also,
Y_2 - y_1
,
conditional on Y_1=y_1
,
has a gamma distribution with shape parameter q
.
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 where
the first column is y_1
and the
second y_2
.
It is necessary that each element of the
vectors y_1
and
y_2-y_1
be positive.
Currently, the fitted value is a matrix with
two columns;
the first column has values ap
for the
marginal mean of y_1
,
while the second column
has values a(p+q)
for the marginal mean of
y_2
(all evaluated at the final iteration).
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.
Balakrishnan, N. and Lai, C.-D. (2009). Continuous Bivariate Distributions, 2nd edition. New York: Springer.
gammaff.mm
,
gamma2
.
shape1 <- exp(1); shape2 <- exp(2); scalepar <- exp(3)
nn <- 1000
mdata <- data.frame(y1 = rgamma(nn, shape1, scale = scalepar),
z2 = rgamma(nn, shape2, scale = scalepar))
mdata <- transform(mdata, y2 = y1 + z2) # z2 \equiv Y2-y1|Y1=y1
fit <- vglm(cbind(y1, y2) ~ 1, bigamma.mckay, mdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
vcov(fit)
colMeans(depvar(fit)) # Check moments
head(fitted(fit), 1)
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