VGAM (version 1.1-14)

leipnik: 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)

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm,

rrvglm

and vgam.

Details

The (transformed) Leipnik distribution has density function $$f(y;\mu,\lambda) = \frac{ \{ y(1-y) \}^{-\frac12}}{ \mbox{Beta}( \frac{\lambda+1}{2}, \frac12 )} \left[ 1 + \frac{(y-\mu)^2 }{y(1-y)} \right]^{ -\frac{\lambda}{2}}$$ 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 \((-1, 1)\).

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(2000, 0.5, 0.1))  # Improper data
fit <- vglm(y ~ 1, leipnik(ilambda = 1), 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