Saddlepoint approximations for the generalized variance and Wilks' statistic

Abstract

Saddlepoint approximations to densities of sums of random variables were introduced by Daniels (1954). These methods were further developed by Barndorff-Nielsen & Cox (1979) to approximate joint and conditional densities for canonical sufficient statistics in exponential family settings. Approximations for the cumulative distributions of sums were developed by Lugannani & Rice (1980) and Robinson (1982), while the approximation of conditional cumulative distribution functions was addressed by Skovgaard (1987). The modifications necessary to apply the Lugannani & Rice and Skovgaard approximations to conditional distributions of canonical sufficient statistics in exponential family settings have been summarized by Davison (1988). Reid (1988) provides an overall review of the saddlepoint literature. In the present paper the Lugannani & Rice, and Skovgaard approximations are used to reproduce the distributions of the generalized variance and Wilks' likelihood ratio statistic in a multivatiate analysis of variance. Representing the degree of freedom in a gamma regression model as an appropriate linear function, the generalized variance distribution arises as the marginal distribution of a canonical sufficient statistic to which the Lugannani & Rice approximation may be applied. The more recent approximations of Fraser (1990) and Barndorff-Nielsen (1990) are exactly the same for this setting. Comparison of its accuracy with the approximation of Hoel (1937) as well as a normal approximation (Johnson & Kotz, 1972, p. 198) shows the Lugannani & Rice approximation to be highly accurate and uniformly better than its competitors. The distribution of Wilks' likelihood ratio statistic has two distinct characterizations in terms of products of independent beta variates (Anderson, 1958, Ch. 8). In conjunction with these characterizations we use Skovgaard's approximation to construct two distinct