The Annals of Statistics

$k$-Statistics and Dispersion Effects in Regression

Peter McCullagh and Daryl Pregibon

Full-text: Open access


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.

Article information

Ann. Statist., Volume 15, Number 1 (1987), 202-219.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 63E30
Secondary: 62J05: Linear regression 62F35: Robustness and adaptive procedures 62N10

Polykay cumulant residual test tensor generalized inverse heterogeneity of variance


McCullagh, Peter; Pregibon, Daryl. $k$-Statistics and Dispersion Effects in Regression. Ann. Statist. 15 (1987), no. 1, 202--219. doi:10.1214/aos/1176350261.

Export citation