7. Probability and Statistics Soviet Essays - Sheynin, Oscar

described, from the viewpoint that interests us, by a definite number of parameters. In thesimplest cases this set is finite. To illustrate: the set of the studied characteristics of abiological individual (stature, weight, volume, etc); the set of the coordinates and impulses ofa certain number of the particles of a physical system. In more complicated cases the set ofthe parameters is infinite, as it occurs for example for the field of the velocities of a turbulentcurrent of liquid; for the field of pressure or temperature of the Earth’s atmosphere, etc. Theexceptional complexity of the processes taking part in such systems compels us to applystatistical methods of research. In following the statistical approach, we consider eachobserved state of the system as a random representative, or specimen selected by chancefrom an abstract general population of the states possible under identical general conditions.We assume that over this general population the random parameters can possess somedistribution of probabilities corresponding to certain conditions usually formulated as ahypothesis. In the simplest cases this will be a multivariate distribution; and, for an infinitenumber of parameters, a distribution of a random function or of a random field in afunctional space.The observed data can be either the registered states of a more or less vast population ofspecimens of the given system (the states comprising a sample from the general population),or only some mean (space or temporal) characteristics of the states of the system. Theinterrelations between the empirical material and the theoretically allowed distribution of thegeneral population constitute the main subject of mathematical statistics. Included problemsare, for example, the fullest and most precise reconstruction of the law of distribution of thegeneral population, given the sample; an adequate check of various hypotheses concerningthis population; an approximate estimation of the parameters and of the theoretical meanscharacterizing the theoretical distribution; an interpretation of various relations anddependences observed in samples; and many other practically and theoretically vital pointsoriginating in the applications of the statistical method.I shall now go over to characterize separate prominent achievements of Soviet scholars insolving the most important problems of mathematical statistics.1. The Theory of the Curves of Distribution. Correlation TheoryThe limit theorems of the theory of probability which determine the applicability, undervery general conditions, of the normal law to sums of independent or almost independentvariables, ensure the suitability of the theoretical model of a normally distributed populationto many concrete problems. Already the early statistical investigations made by Quetelet andthen widely developed by the British Galton – Pearson school ascertained that the normal lawwas rather broadly applicable to biological populations. At the same time, however, it wasalso established that considerable deviations from the usual picture of the normaldistribution, viz., an appreciable skewness and an excess of some empirically observeddistributions, were also possible. To describe mathematically the distributions of such a type,Pearson introduced a system of distribution functions which were solutions of the differentialequationdy x − a(1/y) =2dx b + b x + b0 1 2xand worked out in detail the methods of determining the parameters of the appropriate curvesgiven the empirical data. It occurred that the Pearsonian curves, very diverse in form, wereapplicable for interpolation in a broad class of cases. However, for a long time theirstochastic nature was left unascertained; Pearson’s own substantiation that he provided insome of his writings was patently unsound and led to just criticisms (Yastremsky [1]). The