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

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the increase of the probability of failure with time, was formulated.[...]The values of the frequencies of failures comprise a broken line.Their scatter increases with t, a circumstance connected with a sharpdecrease of the area of insulation, i. e., of the amount of experimentalmaterial.We are interested in the values of probabilities p(t) of a failure of aunit area of insulation aged t during unit time (10 4 working hours,about 1.5 years). For small values of t the amount of experimentalmaterial is large, but p(t) themselves are low, 0.01 – 0.02, so that theirdirect determination through frequencies is fraught with very largeerrors. The mean square deviation of the frequency, µ i /S i , where µ i isthe number of failures during time interval between (i – 1)-th and i-thtime units and S i , the corresponding area of insulation, is known to beequal top( ti)[1 − p( ti)]Siwhere t i =10 4 i hours. For t 1 = 10 5 p(t i ) ≈ 0.02, S i = 200, so thatdeviation is roughly 0.01 or 50% of p(t) itself.Then it is natural to attempt to heighten the precision of determiningp(t) by smoothing since the estimation of this probability then dependson all other experimental data. But then, a statistical model isnecessary here. It is rather natural to consider the observed number offailures (1.2) as random variables with a Poisson distribution.Understandably,Eµ i = S i p(t i ).It is somewhat more difficult to agree that the magnitudes µ i areindependent. Here, however, the following considerations applicableto any rare events will help. Take for example µ 1 , µ 2 . Failure occurringduring the first interval of time influences the behaviour of theinsulation in the second interval, but that action is only restricted to thefailed machines whose portion was small. Having admittedindependence, the mathematical model is completely given although itis connected not with the most convenient normal, but with thePoisson distribution. Then, the variancesvar µ i = Eµ i = S i p(t i )depend on probabilities p(t i ) which we indeed aim to derive. Atransformation to magnitudesv = 2 µ(2.1)iiessentially equalizes the variances and therefore helps.These magnitudes v 1 , v 2 , ..., v n from which we later return tomagnitudes (1.2) are smoothed. The smoothing itself is easy in essence58
ut some mathematical tricks described in detail elsewhere (Belova etal 1965, 1967) are applied.The approximate expressionp(t i ) = p(x i ) ≈ (1/4)[b 0 – 0.1333b 2 + b 2 x 2 ] 2 + 0.35/S i , x i = t i /22,where t i is measured in the selected intervals of time and the last termis necessary for allowing for the systematic error that occurred whentransferring to v i (2.1) should be considered final.Estimates for b 0 and b 2 and their variances arebˆ = 0.225, bˆ = 0.20, var bˆ = 2.12⋅ 10 , var bˆ= 44.5⋅10 .−4 −40 2 0 2The magnitudes 0.35/S i are smoothed by a certain polynomial.Careful statistical work concerning probabilities of failures shouldapply our answer exactly in the provided form. However, it is not vividenough and we have therefore represented it in a simplified way. Aconfidence rectangle for (b 0 , b 2 ) with an 80% coefficient was thereforeindicated with curves p 1 (t), p 2 (t), p 3 (t) added. Curve p 2 (t) provides thebest estimate of the real probability of failure which we are able tooffer. It corresponds to the estimates of b 0 and b 2 . The other curves areobtained if the real point (b 0 , b 2 ) is replaced by the left lower and rightupper vertices of the confidence rectangle respectively. They providean idea about the order of the possible error of curve p 2 (t) but, strictlyspeaking, are not the boundaries of the confidence region for the truecurve. The confidence region for p(t) can be constructed in differentways. Strictly speaking, it is not needed since all the informationapplied for constructing it is summed in the mentionedvariances, var bˆˆ0and var b2.The region between p 1 (t) and p 3 (t) can beconsidered as some approximate (having the adequate order) versionof the confidence region.Various versions of checking our model by statistical tests are offundamental significance. In statistics, no check is exhausting but anumber of well passed tests nevertheless produces a feeling ofcertitude in the results. We will dwell in detail on all these checks.The simplest criterion is the study of the final result. Let us have agood look at the curve representing the dependence sought. Do not theactual data deviate too much? (The maximal among the 22 deviationsis µ 20 = 4 and according to the Poisson formula P{µ 20 ≥ 4} ≈ 0.12.) Initself, this is not especially significant, and for the maximal of 22deviations with only 1/22 ≈ 0.05, it is quite acceptable. [...]It is possible to compile an expression similar to the sum of thesquares of deviations of the experimental data from the smooth curvep 2 (t). But it is better to deal with magnitudes v i (2.1) since theirvariance does not essentially depend on the unknown probabilities p(t i )and is roughly equal to 1. In our case, the variance of the observationsis thus known almost exactly, a circumstance connected with thePoisson distribution, which depends only on one parameter rather thantwo as the normal law does. In general, the test applying the sum of thesquares of deviations also shows that the final curve fits well enough.59
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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,
Page 7 and 8: 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: is applied with P(t) being a polyno
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 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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a linear function is not restricted
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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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