prentice74 {VGAM}R Documentation

Prentice (1974) Log-gamma Distribution

Description

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

Usage

prentice74(llocation = "identity", lscale = "loge", lshape = "identity",
           elocation = list(), escale = list(), eshape = list(),
           ilocation = NULL, iscale = NULL, ishape = NULL, zero = 2:3)

Arguments

llocation, lscale, lshape

Parameter link function applied to the location parameter a, positive scale parameter b and the shape parameter q, respectively. See Links for more choices.

elocation, escale, eshape

List. Extra argument for each of the links. See earg in Links for general information.

ilocation, iscale

Initial value for a and b, respectively. The defaults mean an initial value is determined internally for each.

ishape

Initial value for q. If failure to converge occurs, try some other value. The default means an initial value is determined internally.

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}. See CommonVGAMffArguments for more information.

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 != 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 lgamma3ff.

Special cases: q = 0 is the normal distribution with standard deviation b, q = -1 is the extreme value distribution for maxima, 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.

Value

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

Warning

The special case q = 0 is not handled, therefore estimates of q too close to zero may cause numerical problems.

Note

The notation used here differs from Prentice (1974): alpha = a, sigma = b. Fisher scoring is used.

Author(s)

T. W. Yee

References

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

See Also

lgamma3ff, lgamma, gengamma.

Examples

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

[Package VGAM version 0.8-4 Index]