1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

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Such imagining is a peculiar feature of mathematical statistics, andthere are no clear rules for empirically verifying the results of thetrials.3.4. Convergence in probability. For experimentally checking it 5 along series of secondary trials is required and many samples of size nare needed. The author calls forming many samples the pattern of many series,and the patterns of an extended and a fixed series are now both called the pattern ofone (extended or fixed) series. In mathematical statistics, an ensemble ofsequences of trials is only imagined.3.5. Two competing mathematical models of statistical stability.Thus, the traditional formulation of the limit theorems lack clear rulesfor verifying either the conditions, or conclusions. This is the reasonthat had formerly engendered an illusion, not completely dissociatedfrom, that the laws of large numbers theoretically deduce stability ofmeans from homogeneity of trials. In particular, it followed thatmathematical statistics identifies statistical stability with convergencein probability as studied in the laws of large numbers. The Misesmodel of stability P = lim ω, n → ∞, is not usually mentioned. Theauthor quotes Kolmogorov’s pertinent remark (1956, p. 262):Such considerations can be repeated an unrestricted number oftimes, but it is quite understandable that it will not completely free usfrom the necessity of turning during the last stage to probabilities inthe primitive, rough understanding of that term 6 .To put it otherwise, there is no other way out except turning to thepattern of one series, i. e. to the Mises model of stability offrequencies. If you wish, the Mises definition of probability is exactlythe turn to probabilities in the primitive rough understanding of thatterm. According to common sense, the turn to the last stage should bedone in such a manner that the probabilities of the highest rankincluded in the mathematical model of the given experiments wereindeed actually measured in that experiment. It is apparently difficultto warrant the imagination of probabilities of even one superfluousrank. Nevertheless, such imagination is one of the fundamentals of themethod of mathematical statistics.3.6.1. Postulate of the existence of a distribution of probabilitiesfor the initial random variables. All the considerations inmathematical statistics usually begin by postulating the existence, andsometimes even the concrete type of the distribution of probabilitiesfor unpredictable magnitudes, then the estimation of density orparameters of the objectively existing distribution is demanded. TheFisherian theory of estimation is constructed according to this patternas also the method of maximal likelihood, the theories of confidenceintervals, of order statistics etc 7 . An alternative (see Chapter 2) is toconcentrate on empirical justification of predictions of statisticalstability.The most difficult and interesting problem of empiricallyinvestigating statistical stability is rapidly sped by. Here is Grekova’scritical remark (1976, p. 111) about calculating a confidence intervalwhen the number of trials is small:128
A rather subtle arsenal is developed based on the assumption thatwe know the distribution of probabilities of the random variable (thenormal law). And once more the question emerges: wherefrom indeeddo we know it? And how precisely? And, finally, what is the practicalvalue of the product itself, of the confidence interval? A small numberof trials means small amount of information, and things are bad for us.But, whether the confidence interval will be somewhat longer orshorter, is not so important the less so since the confidence probabilitywas assigned arbitrarily.From my viewpoint, this remark is still a rather mild doubt. We mayadd: Wherefrom and how precisely do we know that, given thisconcrete situation, it is proper at all to discuss distributions ofprobabilities? Suppose, however, that the distribution of probabilitiesof the unpredictable magnitudes under discussion does exist. But then(Grekova 1976), it is not necessary to think highly of the theory ofestimation. Indeed, this theory allows us to extract the maximalamount of information not from sample data in general; the postulateon the type of distribution of probabilities is also introduced. It onlyrepresents reality with some precision at whose empirical estimationthe estimation theory is not at all aimed.And the theory’s conclusions and it itself, generally speaking,changes with the change of that distribution. It would have beennecessary to calculate the vagueness of the sought estimates of theparameters caused by the expected vagueness of the postulateddistribution. Then, the estimation theory extracts the maximal amountof information according to some specific criteria whose practicalvalue is not doubtless. Finally, that theory is based on the postulate ofindependent trials with which, as we saw, not everything was in order.It ought to be stated that the treatises on mathematical statistics do notmiss the opportunity to identify the treatment of observations, thatreally not at all simple discipline, with the scientific approach instatistics. Here is Grekova (1976, p. 112) once more:Mathematical arsenals have some hypnotic property andresearchers are often apt to believe unquestionably their calculations,and the more so the more flowery are their tools [...].In any applied science, a scientific approach presumes first of all acreation of an intuitively convincing empirical foundation. Thecomplication, rigour and cost of the mathematical arsenal should becoordinated with the reliability of the foundation. This pragmatic ruleapplied from long ago is neatly called principle of equal stability of allthe elements of an [applied − Yu. A.] investigation (Grekova 1976, p.111). The theory of estimation hardly satisfies it in due measure.3.6.2. Postulate on the existence of a distribution of probabilitiesfor sample estimates. Imagining many additional samples. Theexistence and sometimes even the type of that distribution ispostulated. Suppose that an experiment according to the pattern ofmany series is carried out. We may only repeat what was said in §3.6.1 concerning the distributions of the initial random variables.129
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Studies in the History of Statistic
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Introduction by CompilerI am presen
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(Lect. Notes Math., No. 1021, 1983,
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sufficiently securely that a carefu
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is energy?) from chapter 4 of Feynm
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demand to apply transfinite numbers
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for stating that Ω consists of ele
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chances to draw a more suitable apa
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Let the space of elementary events
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2.3. Independence. When desiring to
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Eξ = ∑ aipi.Our form of definiti
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absolutely precisely if the pertine
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where x is any real number. If dens
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probability can be coupled with an
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Nowadays we are sure that no indepe
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λ = λ(T)with λ(T) being actually
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(1/B n )(m − A n )instead of the
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along with ξ. For example, if ξ i
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µ( − p0) ÷np0 (1 − p0)nhas an
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distribution of the maximal term |s
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ξ (ω) + ... + ξ (ω)n1n{ω :|
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P{max ξ(t) ≥ x} = 0.01, 0 ≤ t
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1. This example and considerations
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IIV. N. TutubalinTreatment of Obser
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structure of statistical methods, d
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Suppose that we have adopted the pa
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and the variances are inversely pro
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It is interesting therefore to see
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is applied with P(t) being a polyno
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ut some mathematical tricks describ
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It is clear therefore that no speci
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of various groups of machines, and
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nnA(λ) x sin λ t, B(λ) = x cosλ
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of the mathematical model of the Br
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dF(λ) = f (λ) dλ, so that B( t
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usually very little of them. Indeed
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This is the celebrated model of aut
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applications of the theory of stoch
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Page 81 and 82: Reasoning based on common sense and
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Page 85 and 86: IIIV. N. TutubalinThe Boundaries of
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Page 111 and 112: 258 - 82 - 176 cases or 68.5% of al
Page 113 and 114: The Framingham investigation indeed
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Page 117 and 118: What objections can be made? First,
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Page 121 and 122: Kolman E. (1939 Russian), Perversio
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Page 127: The foundations of the Mises approa
Page 131 and 132: 4.3. General remarks on §§ 4.1 an
Page 133 and 134: BibliographyAlimov Yu. I. (1976, 19
Page 135 and 136: processes are now going on in the s
Page 137 and 138: obtaining a deviation from the theo
Page 139 and 140: VIOscar SheyninOn the Bernoulli Law
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1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

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