VGAM (version 1.1-8)

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.

Arguments

lmu, llambda

Link function for the $\mu$ and $\lambda$ parameters. See Links for more choices.

imu, ilambda

Numeric. Optional initial values for $\mu$ and $\lambda$.

Author

T. W. Yee

Details

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