7. Probability and Statistics Soviet Essays - Sheynin, Oscar

observations for its power with regard to a certain class of alternatives to become not lessthan 1/2, the Kolmogorov criterion under the same conditions only demands n 4/5observations. In other words, the limiting efficiency of the former test as compared with thelatter is zero.Non-parametric methods are now in the stage of intense development. The problemsconnected with the estimation of their comparative power and efficiency in differentsituations occurring in practice are far from being solved. A number of important statisticalproblems is not yet covered by non-parametric criteria (for example, the estimation of theagreement between a hypothetically given multidimensional law of distribution and thecorresponding empirically observed distribution; of the discrepancy between twoindependent samples from a multidimensional general population, etc).For a number of practical applications some criteria of the non-parametric typenevertheless demand a preliminary estimate of the parameters and new problemsinsufficiently studied even in the simplest cases (for example, already when checking thenormality of the observed distribution of a given indicator) present themselves here. Inaddition, applications demand specification of asymptotic formulas and compilation of tablesof the distribution functions of tests adapted for finite samples.Many new problems whose solution is very difficult have appeared before non-parametricstatistics in connection with the theory of stochastic processes which is the main channel ofinvestigations stimulated by ever increasing demands made by physics and technology. Inthis comparatively young field of science the possibility of a fruitful application of statisticalmethods, possessing a wider scientific foundation and not restricted by the narrowassumptions of the classical methodology, appears as a significant and progressivephenomenon.Notes1. {The author said nothing about the studies of population, the main field of work of theContinental direction of statistics since the end of the 19 th century (Lexis, Bortkiewicz,Chuprov, Markov).}2. {Already in the 19 th century, statistics became essential for several branches of medicineand psychology, see my papers in Arch. Hist. Ex. Sci., vol. 26, 1982, and Brit. J. Math., Stat.Psychology, vol. 57, 2004.}References1. Kolmogorov, A.N. (1942), Determining the center of scattering and the measure ofprecision given a restricted number of observations. Izvestia Akad. Nauk SSSR, ser. Math.,vol. 6, pp. 3 – 32. Translated in Probability and Statistics. Russian papers of the SovietPeriod. Berlin, 2005.2. Kolmogorov, A.N. (1950), Unbiased estimators. Translated in his Sel. Works, vol. 2.Dordrecht, 1992.3. Kolmogorov, A.N. (1951), % & ... (StatisticalAcceptance Inspection When the Admissible Number of the Accepted Defective Articles IsZero). L.4. Sirazhdinov, S.Kh. (1957), Unbiased estimators of the relative number of accepteddefective articles when applying the method of single sampling. Trudy Inst. Matematiki iMekhaniki Akad. Nauk Uzbek Soviet Socialist Republic, vol. 20, pp. 89 – 100.5. Mikhalevich, V.S. (1956), Sequential Bayesian solutions and the optimal methods ofstatistical acceptance inspection. Teoria Veroiatnostei i Ee Primenenia, vol. 1, 1956, pp. 437– 465.6. Aivazian, S.A. (1959), Comparison of the optimal properties of the Neyman – Pearsonand the Wald criteria. Ibidem, vol. 4, 1959, pp. 86 – 93.