7. Probability and Statistics Soviet Essays - Sheynin, Oscar

for finite values of N, M and R 2 . For M = R precise and rather simple formulas for thedistributions D N,N , D N,N + and for their joint distribution were {also} derived (Gnedenko &Koroliuk [60], Gnedenko & Rvacheva [64]). Koroliuk [6] determined the distributionfunctions of D M,R and D M,R + . For M = pR, assuming that p , he derived the distribution ofthe statistics D N + and D N . Rvacheva [8], also see Gnedenko [79], considered the maximaldiscrepancy between two empirical distributions not on all the axis, but on an assignedstochastic {random?} interval. The works of Ozols [1; 2] adjoin these investigations.In addition to the listed issues, problems concerning the mutual location of two empiricaldistribution functions were also considered. Such, for example, was the problem about thenumber of jumps experienced by the function F M (x) and occurring above F R (x). Gnedenko &Mikhalevich [68; 69] studied the case M = pR, and Paivin (Smirnov’s student) considered thegeneral case ({although} only its limiting outcome). Mikhalevich [3] derived the distributionof the number of intersections of the function F M (x) with the broken liney = F R (x) + zM + RMRfor M = R; also see Gnedenko [79].The test of a hypothesis that a distribution belongs to a given class of distributions presentsa more general problem than that of testing the agreement between empirical data and aprecisely known distribution function. The mentioned class of distributions can, for example,depend in a definite way on a finite number of parameters. Gikhman [13; 16; 17] initiatedsuch investigations by studying these problems for the Kolmogorov and the 2 tests. Atabout the same time Darling (1955) examined similar problems for the latter test.The investigation of the 2 goodness-of-fit test for continuous distributions and anunbounded increase both in the number of observations and in the intervals of the grouping isrelated to the issues under discussion. Tumnanian’s [1] and Gikhman’s [19] findings belongto this direction. We only note that this problem is akin to estimating densities.5. During the last years, a considerable number of studies were devoted to developingstatistical methods of quality control of mass manufactured goods. These investigationsfollowed three directions: empirical studies; development of methods of routine control; andthe same for acceptance inspection. Drawing on these works, Tashkent mathematiciansworked out a draft State standard for acceptance inspection based on single sampling. Suchissues are important for practice, and several conferences held in Moscow, Leningrad, Kievand other cities were devoted to them.Contributions on routine statistical control were mostly of an applied nature; as a rule, theydid not consider general theoretic propositions. The investigations of Gnedenko, Koroliuk,Rvacheva and others (§4) nevertheless assumed these very contributions as a point ofdeparture. In most cases the numerous methods of routine statistical quality control offeredby different researchers regrettably remained without sufficient theoretical foundation. Thus,an analysis of their comparative advantages and economic preferences is still lacking. Fromthe mathematical point of view, the method of grouping proposed by Fein, Gostev & Model[1] is perhaps developed most of all. Romanovsky [117] and Egudin [7] worked out itstheory and Bolshev [1; 2] suggested a simple nomogram for the pertinent calculations.Egudin provided vast tables adapted for practical use and Baiburov [1] with a number ofcollaborators constructed several appropriate devices {?}.A number of scientists investigated problems of acceptance inspection, but we only dwellon some findings. As stated above, Kolmogorov [129] applied the idea of unbiased estimatesfor such inspection. He assumed that one qualitative indicator was inspected after which the