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

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Another group of tests is connected with the choice of the degreeand the number of terms of the approximating polynomial. Here, wealso deal with v i (2.1) and test what happens when they areapproximated by various polynomials up to the third degree inclusive.It is obvious that the polynomial sought includes a free term. Then weadd, in turn, terms of the first, second and third degree. The bestimprovement of approximation is reached when polynomials of thetypec 0 + c 2 t 2 (2.2)are chosen.And now we check that the addition of terms of the first and thirddegree to it does not significantly improve the approximation; fordetails, see Belova et al (1965, 1967). After all these checks webecome sure that applying a polynomial (2.2) we have indeed ascompletely as was possible elicited the determinate component fromthe available data.However, having happily concluded the tests of the hypothesesconnected with the smoothing, we do not at all check the mainhypothesis, that the probability of the failure of a unit area ofinsulation does not depend on the constructive or operationalpeculiarities of the pertinent machine. Indeed, we are only checkingwhether the magnitudes µ i are obeying the Poisson distribution (and, inpart, whether they are independent).However, that distribution also occurs when the probabilities offailures occurring on different areas of the insulation are unequal(provided all the probabilities are sufficiently low). The mostimportant hypothesis of statistical homogeneity of the various unitareas of insulation is yet left unchecked and can not be checked byissuing from the generalized data of Fig. 1 4 . At the same time mostinteresting is exactly the isolation and study of machines with high andlow break-down rates (or a confirmation that all of them have the samerate of failures). We will see now how these problems can be solved.2.3. Check of statistical homogeneity. The most importantcondition of acquiring a statistically homogeneous totality, or, so tosay, the most important mystery of the statistical art consists incarefully selecting the material to be studied. Thus, the data of Fig. 1does not include failures of the insulation occurring because of causes[of various causes of its random damage]. We supposed that suchcauses, although usually called random, are not random in thestochastic sense since they are not statistically stable.The selection of material was made easier by the fact that a failureof a large machine is an extreme event whose causes are thoroughlyinvestigated and duly registered. The most suitable for including afailure into statistical treatment was the formulation local defect ofinsulation. In general, however, all failures were included if an aliencause was not clearly indicated. Failures included into statisticaltreatment composed about a half of all the failures of insulation.When selecting material, the statistician must invariably keep to someprinciple once and for all.60
It is clear therefore that no special significance can be attached tostatistical calculations of reliability. This conclusion is important for aprincipled evaluation of the real meaning of the reliability theory.Now, however, our interest is concentrated on another point, onascertaining whether our thoroughly selected totality was statisticallyhomogeneous. Suppose that practically the derived curve preciselyexpresses the probability of failure, p(t). If the failures of the insulationare mostly due to its local damage, it is logical to assume that a failureof a certain machine does not influence (or little influences) its failureafter repair.But then the total number of failures ξ i during all the operationaltime of a machine is a sum of independent random variables, − thenumber of failures during the first, the second, ... selected intervals oftime. Each term obeys the Poisson distribution, so that the totalnumber of failures also obeys it. The parameter of that distribution formachine i for the (k – 1)-th time interval isλ ik = p(t k )S i ≈ p 2 (t k )S i (2.3)where as before S i is the area of insulation of machine i.Therefore, the parameterλ i = Eξ i (2.4)of the total number of failures for the i-th machine can be calculatedby summing the expressions (2.3) over such t k that are less than thegeneral working time of the pertinent machine. We may thus considerthat the numbers (2.4) are known for all the machines. [...] Thismethod of determining λ i is only valid when statistical homogeneity issupposed, otherwise the computed curve p 2 (t) only provides a generalcharacteristic of the breakdown rate.Some machines will have a higher, other machines, a lower rate, −will have either more or less failures than indicated by the Poisson lawwith parameter calculated according to our rule. So it seems that wehave established the effect to be sought for in order to check violationsof statistical homogeneity. However, the trouble is that it is verydifficult to discern that effect. Indeed, suppose we have determinedthat for a certain machine λ i = 0.1 whereas ξ i = 2. Since2λiλ 1ie −P{ξ i≥ 2} = + ... ≈2 200it would seem that we detected a significant departure from thathomogeneity. But statistics covers several hundred machines, so thatfor one (and even for a few) of them an event with probability 1/200can well happen.There are several possible ways for establishing a useful statisticaltest of homogeneity. One of them is, to apply the Poisson theoremonce more. Consider the total number of machines that experiencedone, two, three, ... failures. We will show that the distribution of61
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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
Page 9 and 10: is energy?) from chapter 4 of Feynm
Page 11 and 12: demand to apply transfinite numbers
Page 13 and 14: for stating that Ω consists of ele
Page 15 and 16: chances to draw a more suitable apa
Page 17 and 18: Let the space of elementary events
Page 19 and 20: 2.3. Independence. When desiring to
Page 21 and 22: Eξ = ∑ aipi.Our form of definiti
Page 23 and 24: absolutely precisely if the pertine
Page 25 and 26: where x is any real number. If dens
Page 27 and 28: probability can be coupled with an
Page 29 and 30: Nowadays we are sure that no indepe
Page 31 and 32: λ = λ(T)with λ(T) being actually
Page 33 and 34: (1/B n )(m − A n )instead of the
Page 35 and 36: along with ξ. For example, if ξ i
Page 37 and 38: µ( − p0) ÷np0 (1 − p0)nhas an
Page 39 and 40: distribution of the maximal term |s
Page 41 and 42: ξ (ω) + ... + ξ (ω)n1n{ω :|
Page 43 and 44: P{max ξ(t) ≥ x} = 0.01, 0 ≤ t
Page 45 and 46: 1. This example and considerations
Page 47 and 48: IIV. N. TutubalinTreatment of Obser
Page 49 and 50: 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: ut some mathematical tricks describ
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 and 102: values of (2.1) realized in the n e
Page 103 and 104: *several dozen. The totality µ ica
Page 105 and 106: Mendelian laws. It is not sufficien
Page 107 and 108: example, the problem of the objecti
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258 - 82 - 176 cases or 68.5% of al
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The Framingham investigation indeed
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or, for discrete observations,IT(ω
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What objections can be made? First,
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eliability and queuing are known to
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Kolman E. (1939 Russian), Perversio
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measurement is provided. Recently,
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which means that sooner or later th
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The foundations of the Mises approa
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A rather subtle arsenal is develope
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4.3. General remarks on §§ 4.1 an
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BibliographyAlimov Yu. I. (1976, 19
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processes are now going on in the s
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obtaining a deviation from the theo
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VIOscar SheyninOn the Bernoulli Law
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1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

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