7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Mikhalevich’s findings by far exceeded the boundaries of the problems of acceptanceinspection.6. Statistical practice often has to decide whether observations considerably divergingfrom the others or from the mean are suitable. Numerous pertinent rules established in theliterature are often groundless and are only being applied out of tradition. Authors, who treatstatistical data, not infrequently reject outlying observations without applying any rules andare therefore usually led to wrong conclusions.Assuming that the initial distribution was normal, Smirnov [20] derived the distribution ofthe deviation of the maximal term of the variational series from the mean normed by theempirical variance. He illustrated his results by a short table of the distribution obtained.Later Grubbs (1950) explicated this result as well; obviously, it remained unnoticed.Bendersky [1], who followed Smirnov, determined the distribution of the absolute value ofthe same deviation, again normed by the empirical variance. Bendersky & Shor [2] publisheda monograph devoted to estimating the anormality of observations complete with examplesworked out in detail, vast tables and criticism of wrong rules. Incidentally, wrongrecommendations had even slipped into the texts of widely used manuals written by eminentauthors. Thus, Romanovsky [106, pp. 25 – 29] advanced a rule based on a misunderstanding:he assumed that the maximal term of a variational series and the mean of the otherobservations were independent.In this section, we shall also touch on some isolated directions of research; in the nearestfuture a few of them will undoubtedly attract considerably more attention. We would like toindicate first of all that the contributions on statistics of dependent trials were uncoordinated.Smirnov [21] recommended a statistic similar to the 2 and derived its limiting distributionfor testing the hypothesis on the constancy of transition probabilities p ij (p ij > 0) in a finiteMarkov chain with (s + 1) states.Linnik [51] offered a method of constructing confidence intervals for the correlationcoefficient in a normal stationary Markov chain under various hypotheses concerning theparameters of a one-dimensional normal distribution.Kolmogorov [114], Boiarsky [10] and others studied the analysis of variance. Bernstein’sinvestigations of the correlation theory, that he explicated in a number of papers, led him tointroducing motions of firm, isogeneous and elastic correlations [35]which proved veryuseful. In a series of papers Sarmanov [7 – 13] developed these ideas. Working on anotheraspect of correlation theory, Mitropolsky [16; 18] fulfilled a number of studies mostlydevoted to correlation equations.7. Statistical practice widely uses tables of the main distributions. In some cases, however,tabulation encounters not only practical difficulties, it often leads to more fundamentalcomplications. In the first place we ought to indicate here the tabulation of functionsdepending on several parameters. Of unquestionable interest is therefore the widespread useof successfully compiled nomograms. Some not numerous attempts of such kind were madein the beginning of the 1930s when A.I. Nekrasov compiled a nomogram for the Studentdistribution function. Three nomograms pertaining to correlation theory were included in vol.1 of the Pearson tables. Elementary nomograms for the normal distribution served asillustrations in the well-known books of Glagolev [24] and Frank [12]. However, onlyErmilov initiated a systematic study of nomographic representation of the formulas ofmathematical statistics. In a lengthy paper he [1] provided nomograms of the density and thedistribution function of the Student law, the 2 and the Fisher distribution {the F-distribution?}. Later he published nomograms of the confidence intervals for estimating bothunknown probabilities by observations and the expectation of the normal law.