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

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It does not exist always even if the pertinent magnitudes areintuitively independent. Statistical independence can only be discussedwith complete justification after establishing statistical stability.Independence of a separate measurement is meaningless. Non-correlationof pairs of magnitudes X 1 (s), ..., X n (s) means that E(X i , X j ) = 0 for i, j = 1, ..., n and i≠ j.2.6. The main problem of the applied theory of probability. Afterheuristically forecasting the initial magnitude V, to predict theoretically somesecondary magnitude, their functions. Forecasting the initial magnitudes is alwaysintuitive.2.7. Limit theorems of the theory of probability. For the central limittheorem (CLT) magnitudes X 1 (s), ..., X n (s) are considered statistically independentfor any n and their scatter around their expectations is supposed to be roughly thesame. For the law of large numbers (LLN) the second demand is dropped and thefirst one weakened so that variance can be even replaced by non-correlation. TheCLT is practically admitted if the LLN takes place intuitively.Quantitative estimates during the proof of the laws of large numbersare only possible by means of the [Bienaymé −] Chebyshev inequalitybut they are rough and inexpedient as compared with the CLT. In theinitial period of the development of the probability theory thefundamental importance of limit theorems had been essentiallyexaggerated which is not completely done away with even now 2 . Thus,sometimes statements are made asserting that statistical stability is dueto the LLN.2.8. The Mises approach. His initial concepts are extremely close to beingexperimental. Instead of stability of the empirical mean he postulates the existence ofE(X) = lim E m (X), m → ∞.The pattern of an extended series is meant here. Particular cases are the definition ofprobability as the limit of frequency and of F(X) being the limit of F m (X). Theconvergence can be understood in different ways.Randomness (that is, unpredictability) does not enter directly, thewhole arsenal of tools is typically mathematical. In similar ways,mathematicians discuss derivatives and integrals rather than velocitiesor specific heat. Transitions to the limit are only the means (ornecessary! expenses) of a rigorous formalization 3 . The Mises approachprovides civil rights in the theory of probability for the knownempirical patterns of treating data dating back to the very foundationsof the natural scientific method with its demand of repeatedreproduction of results.The main feature of the Mises approach consists in dealing witheverything as though considering an experiment. Not surprisingly,expectation is introduced in applications according to his postulateoften without citing Mises.2.9. Comparison with the Kolmogorov axiomatization. TheMises approach most likely can not be included within the boundariesof this axiomatization. The main theoretical problem apparentlyconsists in discovering existence theorems for number sequencesconverging to the given beforehand distribution function. Thisproblem is still only solved for weak convergence (Postnikov 1960).The Mises approach should be specially developed by numbertheoreticmethods unusual for the Kolmogorov axiomatics.126
The foundations of the Mises approach can be quite rigorouslyformed as a clear set of axioms. Contrasting it to the axiomatic methodis wholly based on a misunderstanding.2.10. Conclusion. Unpredictability of repeatedly measured initialmagnitudes is neither necessary, nor sufficient for enlisting the theoryof probability since it does not ensure the initial statistical stability, i.e. the stability of the averaged characteristics of the initial magnitudes,which is the really necessary pertinent condition. Independence oftrials is often presumed, but see § 2.5.I did not have an occasion to enlist officially the notion ofindependence of trials. The introduction of controlled conditions Uinto quantitative notions, formulas or propositions of the theory ofprobability however constructed apparently can not be even hoped for.The introduction of the concept of independence of trials is notrequired by the notions of statistical stability and statisticalindependence of magnitudes. On the contrary, it should be based onthese notions.3. Critical Analysis of the Method of Mathematical StatisticsAccording to one of the usual definitions (Nikitina et al 1972),mathematical statistics studies quantitative relations of massphenomena [...] It is closely linked with the theory of probability. [...]Its methods are universal 4 .3.1. An alternative to the general purpose of mathematicalstatistics. All treatises state that that purpose is to provide a universalnumerical theory of measuring averaged characteristics; to find outwhether a given sample is representative.The possibility of constructing such a theory is doubtful since theprecision and reliability of the initial presumptions can hardly becalculated or justified. Those presumptions are intuitive forecasts ofsome permanences. When adopting them, the alternative is to abstainas much as possible from theoretical considerations, to substantiatetheir likelihood of forecasts by empirical induction. We will discusshow to verify experimentally the typical pronouncements made bymathematical statistics.3.2. Traditional interpretation of limit theorems. [Only the Bernoullitheorem is discussed.] It only deals with one series of observations andapplies two fundamental notions of mathematical statistics,independence of trials and convergence in probability.3.3. Independence of trials. Contrary to what is sometimesasserted, stability of the controlled conditions of the experiments is notsufficient and the conditions U can not at all quantitatively enter thetheory of probability. It is less superficially stated that each trialengenders a random variable so that the independence of the n trials isreduced to the statistical independence of the n variables.However, a trial (a measurement of X) only engenders a realizationof a random variable, the number X(s). Mathematical statistics has noclear rules for empirically verifying independence of trials, fordiscussing an ensemble of such series. The correspondence n trials – nrandom variables means imagining an ensemble of random variables.127
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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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Page 77 and 78: achieved by differentiating because
Page 79 and 80: u(x 1 , x 2 , t 1 , t 2 ) = v(x 1 ,
Page 81 and 82: Reasoning based on common sense and
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Page 85 and 86: IIIV. N. TutubalinThe Boundaries of
Page 87 and 88: periodograms. It occurred that work
Page 89 and 90: at point x = 1. However, preceding
Page 91 and 92: He concludes that since the action
Page 93 and 94: The verification of the truth of a
Page 95 and 96: In the purely scientific sense this
Page 97 and 98: ought to learn at once the simple t
Page 99 and 100: the material world science had inde
Page 101 and 102: values of (2.1) realized in the n e
Page 103 and 104: *several dozen. The totality µ ica
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Page 107 and 108: example, the problem of the objecti
Page 109 and 110: a linear function is not restricted
Page 111 and 112: 258 - 82 - 176 cases or 68.5% of al
Page 113 and 114: The Framingham investigation indeed
Page 115 and 116: or, for discrete observations,IT(ω
Page 117 and 118: What objections can be made? First,
Page 119 and 120: eliability and queuing are known to
Page 121 and 122: Kolman E. (1939 Russian), Perversio
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Page 125: which means that sooner or later th
Page 129 and 130: A rather subtle arsenal is develope
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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