7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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Mikhalevich’s findings by far exceeded the boundaries of the problems of acceptanceinspection.6. Statistical practice often has to decide whether observations considerably divergingfrom the others or from the mean are suitable. Numerous pertinent rules established in theliterature are often groundless and are only being applied out of tradition. Authors, who treatstatistical data, not infrequently reject outlying observations without applying any rules andare therefore usually led to wrong conclusions.Assuming that the initial distribution was normal, Smirnov [20] derived the distribution ofthe deviation of the maximal term of the variational series from the mean normed by theempirical variance. He illustrated his results by a short table of the distribution obtained.Later Grubbs (1950) explicated this result as well; obviously, it remained unnoticed.Bendersky [1], who followed Smirnov, determined the distribution of the absolute value ofthe same deviation, again normed by the empirical variance. Bendersky & Shor [2] publisheda monograph devoted to estimating the anormality of observations complete with examplesworked out in detail, vast tables and criticism of wrong rules. Incidentally, wrongrecommendations had even slipped into the texts of widely used manuals written by eminentauthors. Thus, Romanovsky [106, pp. 25 – 29] advanced a rule based on a misunderstanding:he assumed that the maximal term of a variational series and the mean of the otherobservations were independent.In this section, we shall also touch on some isolated directions of research; in the nearestfuture a few of them will undoubtedly attract considerably more attention. We would like toindicate first of all that the contributions on statistics of dependent trials were uncoordinated.Smirnov [21] recommended a statistic similar to the 2 and derived its limiting distributionfor testing the hypothesis on the constancy of transition probabilities p ij (p ij > 0) in a finiteMarkov chain with (s + 1) states.Linnik [51] offered a method of constructing confidence intervals for the correlationcoefficient in a normal stationary Markov chain under various hypotheses concerning theparameters of a one-dimensional normal distribution.Kolmogorov [114], Boiarsky [10] and others studied the analysis of variance. Bernstein’sinvestigations of the correlation theory, that he explicated in a number of papers, led him tointroducing motions of firm, isogeneous and elastic correlations [35]which proved veryuseful. In a series of papers Sarmanov [7 – 13] developed these ideas. Working on anotheraspect of correlation theory, Mitropolsky [16; 18] fulfilled a number of studies mostlydevoted to correlation equations.7. Statistical practice widely uses tables of the main distributions. In some cases, however,tabulation encounters not only practical difficulties, it often leads to more fundamentalcomplications. In the first place we ought to indicate here the tabulation of functionsdepending on several parameters. Of unquestionable interest is therefore the widespread useof successfully compiled nomograms. Some not numerous attempts of such kind were madein the beginning of the 1930s when A.I. Nekrasov compiled a nomogram for the Studentdistribution function. Three nomograms pertaining to correlation theory were included in vol.1 of the Pearson tables. Elementary nomograms for the normal distribution served asillustrations in the well-known books of Glagolev [24] and Frank [12]. However, onlyErmilov initiated a systematic study of nomographic representation of the formulas ofmathematical statistics. In a lengthy paper he [1] provided nomograms of the density and thedistribution function of the Student law, the 2 and the Fisher distribution {the F-distribution?}. Later he published nomograms of the confidence intervals for estimating bothunknown probabilities by observations and the expectation of the normal law.
At least two more authors offered nomograms for the Student distribution constructed byother methods: M.V. Pentkovsky (doctoral dissertation) and James-Levi [6]. Mitropolskyincluded a number of nomograms in his thorough many-volume course [20, 22].During the 40 years (1917 – 1947) a large number of tables was calculated in the SovietUnion. They were mostly published as natural appendices to appropriate articles ormonographs. A comparatively small number of contributions were devoted to tables as such.From among these we indicate Slutsky’s fundamental work [34] where the author, byapplying a number of clever computational tricks, was able to compile faultless five-placetables of the incomplete function admitting a fair interpolation throughout.Smirnov [8] compiled a table for the Kolmogorov distribution; it was reprinted in the USAand included in a number of educational manuals. We also mention tables of the expectationof the correlation coefficient (Dikovskaia & Sultanova [4]) and numerous useful tablesconcerning the practical use of the various methods of statistical quality control.Notes1. {Notation not explained.}2. {Notation used here as well as in the next few lines insufficiently explained. Neitherwere the three contributions mentioned there, in these lines, helpful. I was only able toperceive that D, unlike D + , was concerned with absolute values of some differences.}12. Joint Bibliography to the Two Preceding ContributionsForeword by TranslatorIn addition to what I noted in my Foreword to the previous Joint Bibliography, I mentionseveral more points. First, in both essays references to joint papers were made in anextraordinary way. Thus, Kolmogorov cited Gelfand [75], himself [157] and Yaglom [25]bearing in mind a single contribution. In this particular case I wrote Gelfand, Kolmogorov &Yaglom [75] (arranging the authors alphabetically) and excluded Kolmogorov [157] andYaglom [25] from this Joint Bibliography. Second, Kolmogorov cited some foreign authorswithout providing an exact reference. Also, in a few instances he referred to Soviet authors inthe text itself and I distinguished these cases by mentioning them in a different way, both inthe translation of his paper and here. Example: Rosanov (1958). Third and last, this time, Ionly provide the titles of books; in other cases, I indicate the pertinent periodical, volumenumber, etc.AbbreviationsAN = Akademia NaukC.r. = C.r. Acad. Sci. ParisDAN = Doklady AN (of the Soviet Union if not stated otherwise)IAN = Izvestia AN of the Soviet Union (ser. Math. if not stated otherwise,or no series at all)IMM = Inst. Math. & Mekh.L = LeningradLGU = Leningrad State Univ.M = MoscowMGU = Moscow State Univ.MIAN = Steklov Math. Inst.MS = Matematich. SbornikSAGU = Srendeaziatsk. (Central Asian) Gosudarstven. Univ. (Tashkent)SSR = Soviet Socialist RepublicTV = Teoria Veroiatnostei i Ee Primenenia
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[3] I bear in mind the well-known p
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successes of physical statistics. B
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classes of independent facts whose
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distribution is a corollary of the
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examine in the first place the curv
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12. According to Bortkiewicz’ ter
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generality, the similarities taking
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one on another, as well as the corr
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is inapplicable because the right s
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Instead, Slutsky introduced new not
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abandoned in August 1936, but it is
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last decades, mathematicians more o
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charged with making the leading ple
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motion and a number of others) are
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phenomena. It is self-evident that
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Such new demands were formulated in
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The addition of independent random
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automatic lathes, etc. Here, the ma
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11. Kolmogorov, A.N. Grundbegriffe
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period 1 and remained, until the ap
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of the analytical tool rather than
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with probability approaching unity,
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logic. The ensuing vagueness in his
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2. Gnedenko, B.V. (1949), On Lobach
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will be sufficient, although not ne
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nlimk = 1P(| k (n) - m k (n) | > H
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favorite classical issue as the gam
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and some quite definite (not depend
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influenced by a construction that a
Page 62 and 63: P ij (1) = p ij (1) , P ij (t) =kP
Page 64 and 65: 4.2d. Bebutov [1; 2] as well as Kry
Page 66 and 67: are yet no limit theorems correspon
Page 68 and 69: described, from the viewpoint that
Page 70 and 71: conditional variance and determined
Page 72 and 73: Romanovsky [45] and Kolmogorov [46]
Page 74 and 75: Let S be the general population wit
Page 76 and 77: Part 1. Russian/Soviet AuthorsAmbar
Page 78 and 79: 2. On necessary and sufficient cond
Page 80 and 81: Gnedenko, B.V., Groshev, A.V. 1. On
Page 82 and 83: 52. ( (Mathematical Principl
Page 84 and 85: Kozuliaev, P.A. 1. Sur la répartit
Page 86 and 87: Obukhov, A.M. 1. Normal correlation
Page 88 and 89: 30. Généralisations d’un théor
Page 90 and 91: 22. Alcune applicazioni dei coeffic
Page 92 and 93: 10. A.N. Kolmogorov. The Theory of
Page 94 and 95: Kuznetsov, Stratonovich & Tikhonov
Page 96 and 97: In the homogeneous case H s t = H t
Page 98 and 99: to such a generalization. He only s
Page 100 and 101: In the particular case of a charact
Page 102 and 103: as it is usual for the modern theor
Page 104 and 105: 1. {The second reference to Pugache
Page 106 and 107: Smirnov, Romanovsky and others made
Page 108 and 109: determined the precise asymptotic c
Page 110 and 111: for finite values of N, M and R 2 .
Page 114 and 115: Uch. Zap. = Uchenye ZapiskiUkr = Uk
Page 116 and 117: Khinchin, A.Ya. 43. Math. Ann. 101,
Page 118 and 119: 7. DAN 115, 1957, 49 - 52.Pinsker,
Page 120 and 121: Anderson, T.W., Darling, D.A. (1952
Page 122 and 123: Statistical problems in radio engin
Page 124 and 125: observations for its power with reg
Page 126 and 127: securing against mistakes (A.N. Kry
Page 128 and 129: of the others, then its distributio
Page 130 and 131: In Kiev, in the 1930s, N.M. Krylov
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7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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