Estimates the two parameters of the McCullagh (1989) distribution by maximum likelihood estimation.
for the theta and nu parameters.
Numeric. Optional initial values for theta and nu. The default is to internally compute them.
The McCullagh (1989) distribution has density function
f(y;theta,nu) = (1-y^2)^(nu-0.5) / [ (1 - 2*theta*y+theta^2)^nu * Beta(nu+0.5, 0.5)]
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.
An object of class
The object is used by modelling functions such as
Convergence may be slow or fail unless the initial values are reasonably
close. If a failure occurs, try assigning the argument
itheta. Figure 1 of McCullagh (1989) gives a broad range of
densities for different values of theta and nu,
and this could be consulted for obtaining reasonable initial values if
all else fails.
T. W. Yee
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).
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