Keeping Moment-Like Sampling Computations Simple

Abstract

This is an attempt to present as simply as possible the best tools we know today for keeping computations simple when dealing with samples from general populations. Such computations seem inevitably to be made in terms of quantities related to moments. We develop here the formal structure and inter-relations of the two systems of multi-index quantities which seem today to be best adapted to statistical use. The occurrence of two systems is, at least in part, related to the appearance in statistical problems of both multiplication and addition of independent variables. Hence the existence of two systems, whose limiting cases are moments (about a fixed point) and cumulants (or semiinvariants). We present interconversion formulas, developing definitions and proving the pairing formulas without reference to any infinite populations, and sparing the use of combinatorial techniques as much as we are able. A few multiplication formulas are given, but for a more complete list the reader is referred to Wishart [10]. It is hoped that this paper can be read on its own, with some reference to applications of these techniques to elementary examples [6] and to the sampling properties of estimated variance components in the analysis of variance [7], [8], [9] as motivation. The author's best thanks go to N. R. Goodman for the checking of certain calculations.