VGAM (version 0.9-4)

mccullagh89: McCullagh (1989) Distribution Family Function

Description

Estimates the two parameters of the McCullagh (1989) distribution by maximum likelihood estimation.

Usage

mccullagh89(ltheta = "rhobit", lnu = logoff(offset = 0.5),
            itheta = NULL, inu = NULL, zero = NULL)

Arguments

ltheta, lnu

Link functions for the $\theta$ and $\nu$ parameters. See Links for general information.

itheta, inu

Numeric. Optional initial values for $\theta$ and $\nu$. The default is to internally compute them.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The default is none of them. If used, choose one value from the set {1,2}.

Value

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

Details

The McCullagh (1989) distribution has density function $$f(y;\theta,\nu) = \frac{ { 1-y^2 }^{\nu-\frac12}} { (1-2\theta y + \theta^2)^{\nu} \mbox{Beta}(\nu+\frac12, \frac12)}$$ where $-1 < y < 1$ and $-1 < \theta < 1$. This distribution is equation (1) in that paper. The parameter $\nu$ satisfies $\nu > -1/2$, therefore the default is to use an log-offset link with offset equal to 0.5, i.e., $\eta_2=\log(\nu+0.5)$. The mean is of $Y$ is $\nu \theta / (1+\nu)$, and these are returned as the fitted values.

This distribution is related to the Leipnik distribution (see Johnson et al. (1995)), is related to ultraspherical functions, and under certain conditions, arises as exit distributions for Brownian motion. Fisher scoring is implemented here and it uses a diagonal matrix so the parameters are globally orthogonal in the Fisher information sense. McCullagh (1989) also states that, to some extent, $\theta$ and $\nu$ have the properties of a location parameter and a precision parameter, respectively.

References

McCullagh, P. (1989) Some statistical properties of a family of continuous univariate distributions. Journal of the American Statistical Association, 84, 125--129.

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

leipnik, rhobit, logoff.

Examples

mdata <- data.frame(y = rnorm(n = 1000, sd = 0.2))  # Limit as theta = 0, nu = Inf
fit <- vglm(y ~ 1, mccullagh89, data = mdata, trace = TRUE)
head(fitted(fit))
with(mdata, mean(y))
summary(fit)
coef(fit, matrix = TRUE)
Coef(fit)