7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Statistical problems in radio engineering concerned with revealing signals against thebackground of interferences and noise constituting a stochastic process gave rise to a vastliterature. The study of these and of many other problems caused by the requirements ofmodern technology (for example, by communication techniques as well as by problems inqueuing theory; in studies of microroughness on the surfaces of articles; in designingreservoirs, etc), leads to the statistics of stochastic processes. This is certainly one of the mosturgent and fruitful, but also of the most difficult areas of modern research. Until now, onlyseparate statistical problems connected with testing the most simple hypotheses for Markovprocesses are more or less thoroughly studied. Thus, tests are constructed for checkingsimple hypotheses about transition probabilities or for studying the order of complication{generalization?} of a chain for some alternative, etc.[3] The Scandinavian mathematicians Grenander and Rosenblatt [7] studied a number ofstatistical problems originating when examining stationary processes; from among these theestimation of the spectral density of a series by means of periodograms should be mentionedin the first instance.The entire field of statistics of stochastic processes requires involved mathematical tools,and research often leads to results unexpected from the viewpoint of usual statistics ofindependent series of observations. Already Slutsky (1880 – 1948), the remarkable Sovietscholar, indicated this fact in his fundamental works devoted both to theoretical problems ofstudying stationary series with a discrete spectrum and especially in his investigations ofconcrete geophysical and geological issues.The main channel of studies stimulated by the requirements of physics and technologynowadays lies exactly in the field of statistics of stochastic processes. The following facttestifies that this field excites interest. In September 1960 a conference on the theory ofprobability, mathematical statistics and their applications, organized by the Soviet andLithuanian academies of sciences and the Vilnius University, was held in Vilnius. And, fromamong 88 reports and communications read out there, 26 were devoted to the applicationsand to problems connected with stochastic processes of various types (for example, the studyof the roughness of the sea and the pitching {rolling? The author did not specify} of ships,design of spectral instruments, design and exploitation of power systems, issues incybernetics, etc).[4] Without attempting to offer any comprehensive idea about the entire variety of theways of development of the modern statistical theory and its numerous applications in thisnote, we restrict our attention in the sequel to considering one direction that took shapeduring the last decades and is known in science as non-parametric statistics. Over the lastyears, distribution-free or non-parametric methods of testing statistical hypotheses werebeing actively developed by Soviet and foreign scientists and today they constitute a sectionof the statistical theory peculiar in its subject-matter and the methods applied.The non-parametric treatment of the issues of hypotheses testing by sampling radicallydiffers from the appropriate classical formulations where it was invariably assumed that thelaws of distribution of the random variables under consideration belonged to some definitefamily of laws depending on a finite number of unknown parameters. Since the functionalnature of these laws was assumed to be known from the very beginning, the aims of statisticswere reduced to determining the most precise and reliable estimates of the parameters giventhe sample, and hypotheses under testing were formulated as some conditions which theunknown parameters had to obey. And the investigations carried out by the British Pearson –Fisher school assumed, often without due justification, strict normality of the randomvariables, Such an assumption essentially simplified mathematical calculations.