VGAM (version 1.1-1)

leipnik: Leipnik Distribution 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

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).

See Also

mccullagh89.

Examples

# 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
# }