7. Probability and Statistics Soviet Essays - Sheynin, Oscar

theoretical merits, especially since he introduced a large number of new concepts and openedup practically important paths of scientific research. The justification and the criticism of hisideas is one of the central problems of the current mathematical statistics. Charlier andChuprov, for example, achieved considerable success here whereas many other statisticiansare continuing Pearson’s practical work definitively lousing touch with probability theory;uncritically applying his formulas, they are replacing science by technique of calculation.The purely theoretical problem of analytically expressing any statistical curve, just as anyproblem in interpolation, can always be solved, and by infinitely many methods. And, owingto the more or less considerable discrepancies allowed by the theory of probability, it is quitepossible, even when only having a small number of arbitrary parameters at our disposal, toobtain a satisfactory theoretical curve. Experience shows that in many cases this can beachieved by applying the Pearsonian curves which depend on four parameters; theoretically,however, in the sense of the corresponding stochastic pattern, these are only justified whenthe deviation from the normal curve is small. It would be interesting therefore to discover thecause of the conformity for those cases in which it actually exists given a large number ofobservations (Bernstein 1926).On the other hand, Bruns’ theory supplemented by Charlier that introduces aperturbational factor into the Gauss or the Poisson function provides a theoretical possibilityfor interpolating any statistical curve. However, for a curve considerably deviating from thenormal curve, a large number of parameters can be necessary, and, moreover, in this case thetheoretical meaning of the perturbational factor becomes unclear. Thus, excluding curvesapproaching in shape the Gauss curve, or the Poisson curve 12 , interpolation of statisticaldistributions is of an empirical nature and provides little help in understanding the essenceand regularities of the phenomena considered.Of a certain interest is therefore the rarely applied method suggested by Fechner 13 andemployed later by Kapteyn and some other authors. It consists in that, by an appropriatechange of the variable, the given statistical curve is transformed into a normal curve. Indeed,we have seen that very diverse patterns of the theory of probability lead to the normaldistribution so that it is natural to expect, and especially in biology, that in many cases whenthe measured variable does not obey the Gauss law, it can in one or another way be expressedas a function of one (or of a few) normal random variable(s). Without restricting our effortsto mechanical interpolation, but groping for, and empirically checking theoretical schemescorresponding to the statistical curves {I omit here a barely understandable phrase}, weshould attempt to come gradually to an integral theory of the studied phenomena. In thisconnection, molecular physics is very instructive and it should serve as a specimen fortheoretical constructions in other branches of statistics.[11] The main causes simplifying the solution of the formulated problems in physics are,first, the hardly restricted possibility of experimentation under precisely determinedconditions 14 . The second favorable circumstance is the enormous number of elements,molecules or electrons, with which physics is dealing. The law of large numbers, whenapplied to bodies of usual size, – that is, to tremendous statistical populations, – thus leads tothose absolutely constant regularities which until recently were being regarded as the onlypossible forms of the laws of nature. Only after physicists had managed to studyexperimentally such phenomena where comparatively small populations of molecules orelectrons were participating, as for example the Brownian motion, and to ascertain that thedeviations foreseen by probability theory actually take place, the statement that physicalbodies were statistical populations of some uniform elements was turned from a hypothesisinto an obvious fact.In addition, most complete are the studies of those phenomena of statistical physics thathave a stationary nature. In other branches of theoretical statistics as well we should therefore