leipnik: Leipnik Distribution Family Function

Description

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

Usage

leipnik(lmu = "logit", llambda = logoff(offset = 1),
        imu = NULL, ilambda = NULL)

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 $\lambda$.

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).

Examples

Run this code

# NOT RUN {
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
# }

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