7. Probability and Statistics Soviet Essays - Sheynin, Oscar

In spite of the variety in the concrete conditions of the origin of issues and problems, thenoticeable widening of their field still admitted a single mathematical interpretation.However, in a number of cases new investigations caused by direct practical requirementswere also of fundamental importance since they fostered a better understanding of thecognitive aims and methods of mathematical statistics.It ought to be noted that exactly the investigations concerning the inspection and rejectionof expensive articles led Wald to the creation of a remarkable teaching of sequential analysisthat did not conform to the previous understanding of the statistical science as a theory ofpurely cognitive estimates made by issuing from an already given data. It occurred that toavoid the loss of a considerable share of information, the compilation of statistical material(or the inspection of the objects of a given batch of produced articles, etc) should be plannedand carried out after accounting for the results already achieved during each {previous} stageof the work. The compilation of the material (the sampling) is discontinued when the dataobtained allow to make a decision ensuring that the probabilities of the possible errors of thefirst and of the second kind when testing a hypothesis, – of the errors measuring the risk of awrong rejection of a correct hypothesis and of an acceptance of a hypothesis that does notreally take place, – do not exceed certain boundaries established beforehand or securing thatthe greatest possible damage in case of a wrong decision be minimal (the minimax principle).[2] The new approach to solving such problems led to another formulation of the mainaims of mathematical statistics that stresses its active part characteristic of the theory of themost beneficial direction of practical activity under conditions of incomplete information onan occurring random process; of the theory of a rational choice from among those possibleensuring the least (in the mean) damage and the best use of the information available. Thispoint of view, that Wald was the first to put forward, proved fruitful and allowed to unite intoa single whole the previously developed sections of mathematical statistics (such as thetheory of estimating parameters and of testing hypotheses) and the new ones, – the theory ofstatistical decisions and sequential analysis.It is interesting that, in essence, this new understanding of the main aims of statisticsmakes use, in a more perfect and prudent way, of the old Bayesian concept [1] that issuedfrom prior distributions. This characteristic feature of the new theory sharply separates itfrom the earlier concepts of Fisher and Neyman who resolutely (although not always withsufficient justification) had cut themselves off from any prior estimates and only issued fromobservational material.In our national literature, along with fruitful investigations of acceptance inspection and ofthe estimation of the relative number of wrongly admitted defective articles, carried out inthe spirit of the new ideas put forward by Kolmogorov [2; 3] and Sirazhdinov [4], veryvaluable findings concerning the optimal methods of quality inspection and ensuring the besteconomic results were due to Mikhalevich [5]. Aivazian [6] showed that the Wald sequentialanalysis allowed to reduce by two or three times the volume of observation as compared withthe Neyman – Pearson optimal classical methodology.Along with research into quality inspection, sequential analysis and the theory of decisionfunctions, studies connected with automatic regulation and various problems in radioengineering constitute a substantial part of the modern statistical literature. Statistical andstochastic methods are being assumed as the basis for solving problems in analyzing andsynthesizing various systems of automatic regulation. The regulation of the process ofautomatic manufacturing, of the work of automatic radars and computers demands anallowance for the continuously originating random perturbations and, consequently, calls forapplying the modern theory of stochastic processes and of the statistical methodology oftreating empirical materials based on that theory.