1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

usually very little of them. Indeed, we may trace the change of someeconomic indicator over decades or at best over a few centuries (as inthe case of the Beveridge series). A year usually means oneobservation (otherwise seasonal periodicity which we should somehowdeal with will interfere, and in general most economic indicators arecalculated on a yearly basis). We therefore have tens or hundreds ofobservations whereas calculations show that for a reliable estimate ofthe correlation function or spectral density we need thousands and tensof thousands of them. Already Kendall (1946) formulated thisconclusion in respect of the former.As a result, mathematicians attempt to attain something by selectingan optimal method of smoothing periodograms, but with a smallnumber of observations this method is generally hopeless. The realapplicability of the theory of stochastic processes is in the spherewhere any number of observations is available. Radio physicists havelong ago developed methods allowing easily and simply to obtainestimates of the spectrum of a stochastic process if unnecessary toeconomize on the number of observations. They apply systems offilters separating bands of frequencies (Monin & Jaglom 1967, pt. 2).3.4. A survey of practical applications. Among the creators of thetheory of stochastic processes who had also dealt with statisticalmaterials we should mention Yale, Slutsky and M. G. Kendall (andmost important are Kolmogorov’s contributions, see below). Thoseworks had appeared even before World War II, that is, when automaticmeans of treating the material were unavailable, and these pioneershad to work with hundreds of observations at the most.Thousands and tens of thousands are needed in the correlationtheory because we are attempting to find out too much, not an estimateof one or a few parameters, but infinitely many magnitudes B(u), u =0, 1, 2, ... i. e. the correlation function (or spectral density, the functionf(λ) for λ taking values on [− π, π]).We can choose another approach for achieving practically effectivemethods of correlation theory when having a small number ofobservations, namely, looking for models depending on a smallnumber of parameters. Slutsky provided one such model, the model ofmoving average. Imagine an infinite sequence of independentidentically distributed random variables... ξ −1 , ξ 0 , ξ 1 , ..., ξ n , ...instead of which we observe the sequence... ς −1 , ς 0, ς 1, .. , ς n, ... (3.6)wheremς = ∑ α ξ .(3.7)n k n−kk = 0In other words, ς n is a sum of some number of independentmagnitudes ξ n−k multiplied by suitable α k . Slutsky modelled the system71