gammahyperbola {VGAM} | R Documentation |
Estimate the parameter of a gamma hyperbola bivariate distribution by maximum likelihood estimation.
gammahyperbola(ltheta = "loglink", itheta = NULL, expected = FALSE)
ltheta |
Link function applied to the (positive) parameter theta.
See |
itheta |
Initial value for the parameter. The default is to estimate it internally. |
expected |
Logical. |
The joint probability density function is given by
f(y1,y2) = exp( -exp(-theta) * y1 / theta - theta * y2)
for theta > 0, y1 > 0, y2 > 1. The random variables Y1 and Y2 are independent. The marginal distribution of Y1 is an exponential distribution with rate parameter exp(-theta)/theta. The marginal distribution of Y2 is an exponential distribution that has been shifted to the right by 1 and with rate parameter theta. The fitted values are stored in a two-column matrix with the marginal means, which are theta * exp(theta) and 1 + 1/theta.
The default algorithm is Newton-Raphson because Fisher scoring tends to be much slower for this distribution.
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.
T. W. Yee
Reid, N. (2003) Asymptotics and the theory of inference. Annals of Statistics, 31, 1695–1731.
gdata <- data.frame(x2 = runif(nn <- 1000)) gdata <- transform(gdata, theta = exp(-2 + x2)) gdata <- transform(gdata, y1 = rexp(nn, rate = exp(-theta)/theta), y2 = rexp(nn, rate = theta) + 1) fit <- vglm(cbind(y1, y2) ~ x2, gammahyperbola(expected = TRUE), data = gdata) coef(fit, matrix = TRUE) Coef(fit) head(fitted(fit)) summary(fit)