Prentice (1974) Log-gamma Distribution

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

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"))

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.

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.

imethod, zero

See CommonVGAMffArguments for information.

glocation.mux, gscale.mux, gshape, probs.y

See CommonVGAMffArguments for 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 \(q \ne 0\), 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.

Value

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

References

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

Warning

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

Author

T. W. Yee

Note

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

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)
#> Iteration 1: loglikelihood = -2586.3162
#> Iteration 2: loglikelihood = -2584.6452
#> Iteration 3: loglikelihood = -2584.6286
#> Iteration 4: loglikelihood = -2584.6285
#> Iteration 5: loglikelihood = -2584.6285
coef(fit, matrix = TRUE)  # Note the coefficients for location
#>              location loglink(scale)     shape
#> (Intercept) -2.777286       1.010882 0.9879914
#> x2           2.200447       0.000000 0.0000000