Estimate the parameter of a gamma hyperbola bivariate distribution by maximum likelihood estimation.
Link function applied to the (positive) parameter theta.
Initial value for the parameter. The default is to estimate it internally.
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
The object is used by modelling functions such as
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.
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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)
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