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

Kendall did that even before Moran’s work (1954) appeared. Herestricted his attention to such values of the parameters a and b informula (3.8) which determine a stationary process, and he mostlyworked with series from economics. Such series rarely oscillate aroundone level creating a stationary process. They usually have a tendency,a trend. The production of electrical energy, say, increasesexponentially and therefore has a linear trend when described on alogarithmic scale. The problem consists in describing the deviationsduring different years from the general tendency.Kendall thought it possible to determine the trend by some method(but certainly not by naked eye which is too subjective for a rigorousstatistical school) and to subtract it. This additionally complicates thestatistical structure of the remaining deviations, but there is nothing tobe done about it. Exactly such deviations as though forming astationary process were studied by the method of autoregression.It is difficult to pronounce a definite opinion about his results. Insome cases the statistical tests were happily passed, but not in othercases. May we consider that success was really achieved in thoseformer or should we explain it only by the small number ofobservations? And no explanation is known why, for example, themodel of autoregression with the trend being eliminated does not suitthe series of the cost of wheat but suits the total head of sheep. Nodecisive success in treating economic series was thus achieved.Kendall (1946) investigated the process of autoregressionconstructed according to equation (3.8) by means of tables of randomnumbers; the longest of the modelled series had 480 terms. Inconcluding, let us have a look at the empirical estimate of a correlationfunction (Fig. 3, dotted line). See how much the estimate differs fromthe real values (continuous line) and fades considerably slower thanthe real function.Hannan (1960) published an estimate of the spectral density ofKendall’s series. The graphs of the theoretical density and its variousestimates are shown on Fig. 4. It is seen that they are pretty littlesimilar to the true density. In particular, the later takes a maximalvalue near point λ = π/5 whereas the maximal values of all theestimates are at point λ = π/15.An unaccustomed eye can imagine that small values of the spectraldensity are estimated well enough, but nothing of the sort is reallytaking place. The relative error is here just as great as in the left side ofthe graph, i. e., as for large values of the density. We see that thecorrelation theory, created by the founders of the theory of stochasticprocesses for treating discrete observational series, such as the numberof sunspots in various years or the values of economic indicatorsexactly in those cases did not attain undoubted success.The idea of a mathematical description of wavy processesencountered the practical difficulty in that any proper estimation of thecorrelation function demands not tens or hundreds of separateobservations, but (Kendall 1946) tens and hundreds of pertinent waveswhich means thousands and tens of thousands observations. On theother hand, parametric models such as the model of autoregression hadnot been convincingly statistically confirmed. Consequently, the74