k-statistics and dispersion effects in regression. (English) Zbl 0659.62025

McCullagh, Peter; Pregibon, Daryl

Ann. Stat. 15, 202-219 (1987).

By the term k-statistic or polykay, we mean an unbiased estimate of a cumulant or product of cumulants. Two sets of unbiased estimates are given for the case where the mean response, E(Y), depends linearly on known covariates x. The k’s are symmetric functions of the least-squares residuals and have previously been discussed by F. J. Anscombe [Proc. 4th Berkeley Sympos. Math. Statist. Probab. 1, 1-36 (1961; Zbl 0101.357)]. The \(\ell 's\) are optimal in the sense of having minimum variance under the ideal assumption of normality.
The emphasis here on computability leads to the algebraic inversion of direct product matrices of order \(n^ 3\times n^ 3\) and \(n^ 4\times n^ 4\), a computation that is rarely feasible numerically, even on the fastest computers. This algebra leads to simple straightforward formulae for all statistics up to degree four. Conditions are given under which the k’s are nearly or asymptotically optimal in the sense of being asymptotically equivalent to the corresponding \(\ell 's.\)
A small-scale simulation study provides a comparison between these statistics for finite n. An application to detecting heterogeneity of variance, avoiding the assumption of normality, is given. A new test statistic for detecting systematic dispersion effects is introduced and compared to existing ones. Two examples illustrate the methodology.

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
62J05	Linear regression; mixed models
15A09	Theory of matrix inversion and generalized inverses
62F35	Robustness and adaptive procedures (parametric inference)
62N05	Reliability and life testing

Keywords:

tensors; generalized inverses; residual test; weighted regression; k- statistic; polykay; cumulant; product of cumulants; unbiased estimates; symmetric functions of the least-squares residuals; inversion of direct product matrices; simulation study; detecting heterogeneity of variance; detecting systematic dispersion effects

Citations:

Zbl 0101.357