Leipnik Regression Family Function

Description

Estimates the two parameters of a (transformed) Leipnik distribution by maximum likelihood estimation.

Usage

leipnik(lmu = "logitlink", llambda = logofflink(offset = 1),
        imu = NULL, ilambda = NULL)

Arguments

Details

The (transformed) Leipnik distribution has density function

f(y;mu,lambda) = (y(1-y))^(-1/2) * (1 + (y-mu)^2 / (y*(1-y)))^(-lambda/2) / Beta((lambda+1)/2, 1/2)

where 0 < y < 1 and lambda > -1. The mean is mu (returned as the fitted values) and the variance is 1/lambda.

Jorgensen (1997) calls the above the transformed Leipnik distribution, and if y = (x+1)/2 and mu = (theta+1)/2, then the distribution of X as a function of x and theta is known as the the (untransformed) Leipnik distribution. Here, both x and theta are in (-1, 1).

Value

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

Note

Convergence may be slow or fail. Until better initial value estimates are forthcoming try assigning the argument ilambda some numerical value if it fails to converge. Currently, Newton-Raphson is implemented, not Fisher scoring. Currently, this family function probably only really works for intercept-only models, i.e., y ~ 1 in the formula.

Author(s)

T. W. Yee

References

Jorgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd edition, Volume 2, New York: Wiley. (pages 612–617).

See Also

mccullagh89.

Examples

ldata <- data.frame(y = rnorm(n = 2000, mean = 0.5, sd = 0.1))  # Not proper data
fit <- vglm(y ~ 1, leipnik(ilambda = 1), data = ldata, trace = TRUE)
head(fitted(fit))
with(ldata, mean(y))
summary(fit)
coef(fit, matrix = TRUE)
Coef(fit)

sum(weights(fit))  # Sum of the prior weights
sum(weights(fit, type = "work"))  # Sum of the working weights