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

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conclusions are derived in a purely mathematical way and thereforecertain. However, on the whole everything depends on the model.Statistical methods are as certain (not more or less) as the conclusionsof other sciences applying mathematical means, for example physics,astronomy or strength of material. In practical problems these sciencescan provide guiding lines but can not guarantee that we have correctlyapplied them.1.3. Model of trend with an error. In a mathematical model of anobservational series something is always determinate and somethingrandom. We will consider a model in which that seriesx 1 , x 2 , ..., x nis given by formulax i = f(t i ) + δ i . (1.7)Here t i is the value of some determinate variable specifying the i-thexperiment, f(t), some determinate function (the trend) and δ i , arandom variable usually called the error of that experiment. Thissituation means that the Lord determined the true dependence by f(t)so that we should have observed f(t i ) in experiment i, but that the devilinserted the error δ i .For example, f(t) can represent one or another coordinate of anobject in space as dependent on time, and x i is our measurement of thatcoordinate at moment t i . The devil’s interference δ i can certainly bedeterminate, random or generally of an indeterminate nature. Thus, theobserved x i can be corrupted by a systematic error so that Eδ i is notnecessarily zero. We may assume that Eδ i = C and does not depend oni but it is also possible to consider Eδ i = φ(t i ) is a function of t i . Stillworse will happen if Eδ i depends on a variable u i which we can notcheck. In neither of those cases statistical treatment can eliminate theerrors.However, a sufficiently thorough planning of the observations canallow us to hope that the errors will be purely random in the sense thatstatistical homogeneity is maintained and there is no systematic shift:Eδ i = 0. More precisely, the systematic error will be sufficiently smalland can be neglected. Such situations indeed comprise the scope of thestatistical methods.After recalling what was said in § 1.2 it becomes clear that mostsimple statistical assumptions should be imposed on the errors δ i . Mostoften these errors are supposed to be independent and identicallydistributed. Normality is also usually assumed. Only one of theirdeviations from the model of sample was brought into use: it issometimes thought that their variances are not equal to one another butproportional to numbers assigned according to some considerations.Or, it is assumed that such numbers w i called weights of observationsare known thatw 1 var δ 1 = w 2 var δ 2 = ... = w n var δ n = σ 252
and the variances are inversely proportional to the weights2σvar δ = .iw iI have described the assumptions imposed on the randomcomponent of our observations. Now I pass to their determinatecomponent f(t) otherwise called trend.The most simple and classical case consists in that the function f(t)is of a quite definite class but depends on some unknown parametersc 1 , c 2 , ..., c k :f(t) = F(t, c 1 , c 2 , ..., c k ) (1.8)where the function F is given by a known formula or an algorithm ofcalculation. For example, in case of the motion of an object in spacethose parameters can be understood as its coordinates and velocities atany definite moment; other, more opportune parameters can also beintroduced.Then any coordinate f(t) will be uniquely determined by theparameters and the Newtonian laws of motion (if that object has noengine). The problem consists in determining estimates of theparameters c ˆigiven observations (1.7). It is solved by the Gaussianmethod of least squares: the estimates are determined in such a waythat the minimal value of the functionn∑i=1[ x − F( t ; c ,..., c ]i i 1 k2of c i will be attained at point ( cˆ1,..., c ˆk).More often, however, is the case in which the real dependence f(t) isunknown. Here also the equality (1.8) is applied but the function F ischosen more or less arbitrarily. Thus, a polynomial might be chosenand the method of least squares once more applied.Such a non-classical situation when f(t) is not known beforehanddemands a more detailed analysis, see a concrete example in the nextChapter. Here, however, we describe an absolutely different modelalso applied for statistically treating observational series.1.4. Model of a stochastic process. The main attention is turned tothe isolation of the determinate component, the trend f(t). The valuesthemselves, x i , of the observational series (1.8) are not random;random are only the additional magnitudes δ i considered as errors,noise, and generally the devil’s machinations. Another approach ispossible with randomness being considered the main property of theseries under study which we now denote byξ 1 , ξ 2 , ..., ξ n . (1.9)Here, the most simple model consists in treating that set as arealization of an n-dimensional random variable. Such a model can beuseful if the experiment providing it can be repeated many times over,53
Page 1 and 2: Studies in the History of Statistic
Page 3 and 4: Introduction by CompilerI am presen
Page 5 and 6: (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: Suppose that we have adopted the pa
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 and 102: values of (2.1) realized in the n e
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*several dozen. The totality µ ica
Page 105 and 106:
Mendelian laws. It is not sufficien
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example, the problem of the objecti
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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
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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
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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