ON ACCURACY OF IMPROVED χ2χ^2χ2 APPROXIMATIONS (original) (raw)

Transformations with Improved Chi-Squared Approximations

Journal of Multivariate Analysis, 2000

Suppose that a nonnegative statistic T is asymptotically distributed as a chisquared distribution with f degrees of freedom, / 2 f , as a positive number n tends to infinity. Bartlett correction T was originally proposed so that its mean is coincident with the one of / 2 f up to the order O(n &1). For log-likelihood ratio statistics, many authors have shown that the Bartlett corrections are asymptotically distributed as / 2 f up to O(n &1), or with errors of terms of O(n &2). Bartlett-type corrections are an extension of Bartlett corrections to other statistics than log-likelihood ratio statistics. These corrections have been constructed by using their asymptotic expansions up to O(n &1). The purpose of the present paper is to propose some monotone transformations so that the first two moments of transformed statistics are coincident with the ones of / 2 f up to O(n &1). It may be noted that the proposed transformations can be applied to a wide class of statistics whether their asymptotic expansions are available or not. A numerical study of some test statistics that are not a log-likelihood ratio statistic is discribed. It is shown that the proposed transformations of these statistics give a larger improvement to the chi-squared approximation than do the Bartlett corrections. Further, it is seen that the proposed approximations are comparable with the approximation based on an Edgeworth expansion.

On Accuracy of Improved �2-APPROXIMATIONS

2001

For a statistic S whose distribution can be approximated by χ 2distributions, there is a considerable interest in constructing improved χ 2approximations. A typical approach is to consider a transformation T = T (S) based on the Bartlett correction or a Bartlett type correction. In this paper we consider two cases in which S is expressed as a scale mixture of a χ 2variate or the distribution of S allows an asymptotic expansion in terms of χ 2-distributions. For these statistics, we give sufficient conditions for T to have an improved χ 2-approximation. Furthermore, we present a method for obtaining its error bound.

Approximation Theorems of Mathematical Statistics

This book covers a broad range of limit theorems useful in mathematical statistics, along with methods of proof and techniques of application. The manipulation of "probability" theorems to obtain "statistical" theorems is emphasized. It is hoped that, besides a knowledge of these basic statistical theorems, an appreciation on the instrumental role of probability theory and a perspective on practical needs for its further development may be gained.

An approximation to the F distribution using the chi-square distribution

Computational Statistics & Data Analysis, 2002

For the cumulative distribution function (c.d.f.) of the F distribution, F(x; k; n), with associated degrees of freedom, k and n, a shrinking factor approximation (SFA), G( kx; k), is proposed for large n and any ÿxed k, where G(x; k) is the chi-square c.d.f. with degrees of freedom, k, and = (kx; n) is the shrinking factor. Numerical analysis indicates that for n=k ¿ 3, approximation accuracy of the SFA is to the fourth decimal place for most small values of k. This is a substantial improvement on the accuracy that is achievable using the normal, ordinary chi-square, and Sche à e-Tukey approximations. In addition, it is shown that the theoretical approximation error of the SFA, |F(x; k; n) − G( kx; k)|, is O(1=n 2 ) uniformly over x.