VGAM (version 1.1-14)

prentice74: Prentice (1974) Log-gamma Distribution

Description

Estimation of a 3-parameter log-gamma distribution described by Prentice (1974).

Usage

prentice74(llocation = "identitylink", lscale = "loglink",
           lshape = "identitylink", ilocation = NULL, iscale = NULL,
           ishape = NULL, imethod = 1,
           glocation.mux = exp((-4:4)/2), gscale.mux = exp((-4:4)/2),
           gshape = qt(ppoints(6), df = 1), probs.y = 0.3,
           zero = c("scale", "shape"))

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Details

The probability density function is given by $$f(y;a,b,q) = |q|\,\exp(w/q^2 - e^w) / (b \, \Gamma(1/q^2)),$$ for shape parameter \(q \ne 0\), positive scale parameter \(b > 0\), location parameter \(a\), and all real \(y\). Here, \(w = (y-a)q/b+\psi(1/q^2)\) where \(\psi\) is the digamma function, digamma. The mean of \(Y\) is \(a\) (returned as the fitted values). This is a different parameterization compared to lgamma3.

Special cases: \(q = 0\) is the normal distribution with standard deviation \(b\), \(q = -1\) is the extreme value distribution for maximums, \(q = 1\) is the extreme value distribution for minima (Weibull). If \(q > 0\) then the distribution is left skew, else \(q < 0\) is right skew.

References

Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika, 61, 539--544.

See Also

lgamma3, lgamma, gengamma.stacy.

Examples

pdata <- data.frame(x2 = runif(nn <- 1000))
pdata <- transform(pdata, loc = -1 + 2*x2, Scale = exp(1))
pdata <- transform(pdata, y = rlgamma(nn, loc = loc, scale = Scale, shape = 1))
fit <- vglm(y ~ x2, prentice74(zero = 2:3), data = pdata, trace = TRUE)
coef(fit, matrix = TRUE)  # Note the coefficients for location