Abstract
This article provides a review of the saddlepoint approximation for a M-statistic of a sample of nonnegative random variables with fixed sum. The sample vector follows the multinomial, the multivariate hypergeometric, the multivariate Polya or the Dirichlet distributions. The main objective is to provide a complete presentation in terms of a single and unambiguous notation of the common mathematical framework of these four situations: the simplex sample space and the underlying general urn model. Some important applications are reviewed and special attention is given to recent applications to models of circular data. Some novel applications are developed and studied numerically.
Highlights
The topic of this article is a saddlepoint approximation to the distribution of the M-statistic Tn, precisely Tn (Y1, . . . , Yn ), which is the implicit solution with respect to (w.r.t.) t of n∑ ξ j (Yj ; t) = 0, (1)j =1 where the function ξ j : R+ × R → R is continuous, decreasing in its second argument, for j = 1, . . . , n, R+ = [0, ∞), and where the random variables Y1, . . . , Yn are nonnegative, dependent and satisfy ∑nj=1 Yj = k, for some fixed k > 0
Example 4 reviews the application of the saddlepoint approximation to the bootstrap distribution of the M-statistic in Equation (1)
The saddlepoint approximation combined with the Multivariate hypergeometric—conditional binomial (MH-B) representation can be applied for approximating the distribution of the M-statistic in Equation (1) in finite population sampling, viz
Summary
The topic of this article is a saddlepoint approximation to the distribution of the M-statistic Tn , precisely Tn (Y1 , . . . , Yn ), which is the implicit solution with respect to (w.r.t.) t of n. This article presents the conditional saddlepoint approximation from the general perspective of the urn sampling model. It updates the previous reviews by presenting additional recent important examples It gives a general reformulation with a consistent and homogeneous notation, that corresponds to a single underlying mathematical model (viz., the urn model and the simplex sample space). It includes new important examples and new numerical comparisons. Yn ) follows the multinomial distribution, given in Equation (3), Good [12] proposed a specific saddlepoint approximation for Mn. This article has the following structure.
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