$k$-Statistics and Dispersion Effects in Regression

Abstract

By the term $k$-statistic or polykay, we mean an unbiased estimate of a cumulant or product of cumulants [Fisher (1929) and Tukey (1950), 1956)]. In this paper, 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 Anscombe (1961; 1981, Appendix 2). The $l$'s are optimal in the sense of having minimum variance under the ideal assumption of normality [Pukelsheim (1980)]. 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 $l$'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.