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

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values of (2.5) are essentially less than those of the Kolmogorovstatistic (2.4). Therefore, when applying the Kolmogorov distributionfor (2.5), we will widen the boundaries of the confidence region andthus admit the hypothesis of normality more often than proper.And so, the careful practical application of the Kolmogorov test inthe most elementary (and therefore most common) situation isimpossible. That criterion helps in those cases when many suchexamples are available which were already tested by some statisticalcriteria and we wish to secure a general point of view concerning theirnumerous applications.Let us pass now to the essence of the problem on the confirmationof the Mendelian laws. Here is the classical situation. Some indicationhas two alleles, A (dominant) and a (recessive). Two pure lines withgenotypes AA and aa are taken and compulsorily crossed. A hybridwith genotype Aa emerges with its phenotype corresponding toindication A. Then a second generation is obtained under free crossing.When admitting the hypothesis of absolute randomness of thecombinations of the gametes, the probability of the occurrence ofgenotype aa is 1/4. Only individuals with genotype aa revealindication a in their phenotype so that the probability of its occurrenceis also 1/4. And so, if there will be n individuals in the secondgeneration, the number of occurrences of indication a in the phenotypemay be considered as the number of successes µ in n Bernoulli trialswith probability of success p = 1/4.This is the simplest case of the Mendelian law. Vast experimentalmaterial had been collected up to 1940 from which it was seen that inmany cases such a simplest law was indeed obeyed. Essentialdeviations (perhaps connected with a differing survivorship ofindividuals of different genotypes and other causes) was also revealed.The school of Lyssenko had been attempting to prove that that lawwas not working. To attain that aim, experiments were carried out, inparticular by Ermolaeva (1939). They were peculiar in that thematerial was considered not from all the individuals of the secondgeneration taken together, but separately for families. It is better toexplain the meaning of that term by an example. In experiments withtomatoes a family is consisting of all the plants of the secondgeneration grown in the same box. Each box is sown with seeds takenfrom the fruit of exactly one plant of the first generation. Theseparation into families occurs quite naturally.However, Kolmogorov (see above) showed that Ermolaeva’s mostnumerous series of experiments can be explained exactly by the mostelementary Mendel model. Suppose that for k families numbering n 1 ,n 1 , ..., n k the number of manifested recessive alleles was µ 1 , µ 2 , ..., µ k ,then the classical De Moivre – Laplace theorem [proving that thebinomial law tended to normality] leads to the normed magnitudes* µi− nip 1 3µi= , p = , q = 1− p =n pq 4 4ihaving approximately the standard normal distribution N(0, 1); theprecision of approximation is quite sufficient for n i of the order of102
*several dozen. The totality µ ican thus be considered (if theMendelian model is valid) a sample with theoretical distribution beingthe standard normal law.Kolmogorov studied two most numerous series of Ermolaeva’sexperiments and respectively two samples (2.6) with 98 and 123observations. [...] He obtained λ = 0.82 and 0.75. The probability of abetter fit (a lesser λ) was 0.49 and − 0.37 so that those values of λ werequite satisfactory.A purely statistical investigation thus changed the results: an allegedrefutation of the Mendelian laws became their essential confirmation.Apart from the opponents of the Mendel theory Kolmogorov alsomentioned the work of his followers, Enin (1939) in particular. He didnot subject that paper to a detailed analysis, but indicated that theagreement with the main model of Bernoulli trials was too good (thefrequencies concerning separate families deviated from p = 1/4 lessthan it should have occurred according to the main model of Bernoullitrials). A detailed analysis is instructive from many viewpoints and Iam therefore providing Enin’s main results.He considers the segregation of the tomato hybrids according todiffering leaves: normal and potato-like. His results are separated intotwo groups depending on the time of sowing the seeds of the hybridplants in the hothouse (February or April). [...]All the material except one observation is shown on Fig. 3. Weought to decide now what kind of statistical treatment is needed. Inapplied mathematical statistics the application of each given statisticaltest is objective, [...] but which criteria should be chosen is anessentially subjective question. The answer depends on whichsingularities of the data seem suspicious and the statistician more orless adequately converts this impression into statistical tests. There areno common rules, we can only discuss examples.The matter is that in principle any given result of observations isunlikely (and in our present case of a continuous law of distributionthe probability of any concrete result is simply equal to zero).Therefore, a criterion can also be found that will reject any hypothesisconsidered in any circumstances. We ought not to be here superdiligentand only admit criteria having a substantial sense suitable forthe concrete natural scientific problem. On the other hand, if notwishing to reject some tested hypothesis, it will be usually possible tochoose such criteria that will not do that. Here, we are alreadyspeaking about the honesty of the statistician.Concerning the material presented on Fig. 3, we first turn ourattention to the empirical function for the first series of observations. Itis situated completely above the theoretical function and in general isquite well smoothed by some straight line (dotted on the Figure)almost parallel to the theoretical. The entire difference is some shift tothe left. Since we deal with a shift (we see it perfectly well, but do notknow whether it is significant or not), we ought to apply the test basedon the sample mean. It is equal to – 0.64 and its variance is1/ 11 ≈ 0.30; to remind, the tested hypothesis concerns the standardnormal distribution for the values of µ*. The deviation exceeds two103
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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
Page 51 and 52: Suppose that we have adopted the pa
Page 53 and 54: and the variances are inversely pro
Page 55 and 56: It is interesting therefore to see
Page 57 and 58: is applied with P(t) being a polyno
Page 59 and 60: ut some mathematical tricks describ
Page 61 and 62: It is clear therefore that no speci
Page 63 and 64: of various groups of machines, and
Page 65 and 66: nnA(λ) x sin λ t, B(λ) = x cosλ
Page 67 and 68: of the mathematical model of the Br
Page 69 and 70: dF(λ) = f (λ) dλ, so that B( t
Page 71 and 72: usually very little of them. Indeed
Page 73 and 74: This is the celebrated model of aut
Page 75 and 76: applications of the theory of stoch
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
Page 83 and 84: answering that question is extremel
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: values of (2.1) realized in the n e
Page 105 and 106: Mendelian laws. It is not sufficien
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
Page 123 and 124: measurement is provided. Recently,
Page 125 and 126: which means that sooner or later th
Page 127 and 128: The foundations of the Mises approa
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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