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

More documents

Recommendations

Info

IVYu. I. AlimovAn Alternative to the Method of Mathematical StatisticsAlternativa Methodu Matematicheskoi Statistiki. Moscow, 1980IntroductionBoth mathematicians and those who have been applyingmathematics are often recently expressing their concern that in manyinstances mathematical models noticeably alienate from reality. As aconsequence, the work of highly qualified specialists and valuablecomputer time is used with insufficient effect. Criticism, occasionallyvery sharp, of this situation is seen ever oftener in papers andmonographs for specialists and in textbooks and popular scientificeditions, see for example Blekhman et al (1976); Grekova (1976);Venikov (1978); Vysotsky (1979). It is indicative that a paper of D.Schwarz called On the pernicious influence of mathematics on scienceis didactically quoted in Venikov (1978).In particular, models offered by mathematical statistics are oftenremote from reality. Tutubalin’s booklets [i − iii] are devoted to theconditions and boundaries of the applicability of stochastic methods,and much attention is shown to such problems in his textbook (1972).With respect to its restrictive direction, this booklet adjoins thosepublications. I stress at once that my contribution is not at all opposedto statistics as such.I understand statistics as any calculation of means or other combinedtreatment of experimental data aiming at providing their predictableintegral characteristics. It is assumed that these will be later measuredfor future similar experimental data so that the correctness of thestatistical forecast will be actually checked.I am not at all against the use of mathematics in statistics either;otherwise, the latter is simply unthinkable so that below I am treatingmathematical statistics. Choose any pertinent treatise and you will beeasily convinced that by no means any application of mathematics instatistics is understood as mathematical statistics. After attentivelylooking, it is seen that mathematical statistics is a very specificdiscipline possessing its own peculiar method whose distinctivefeature is the conjecturing of exactly one storey of probabilities calledconfidence probabilities or levels of significance above those reallymeasured in an experiment. It is possible to disagree with such aspecific approach.Mathematics can be applied in statistics in a manner somewhatdifferent from what is prescribed by mathematical statistics.In practice, the principles of statistically treating experimental datawhich are being applied for a long time now have nothing in commonwith confidence probability and are therefore alien to the foundation ofmathematical statistics. We find for example that [a certain magnitude]is equal to 0.0011609 ± 0.000024. Here, only the maximal error of the122
measurement is provided. Recently, physicists have sometimes begunto indicate instead the mean square error of measuring the last digits ofthe experimental result, usually in brackets; for example, the velocityof light in vacuum is [...]. Essential here is that unlike confidenceprobabilities of mathematical statistics, the maximal and the meansquare error were actually measured.For many years, mathematical statistics has been activelypropagandized, but still perhaps even nowadays physicists will beunable to refrain from smiling had we told them, say, that after treatingthe observations of the velocity of light, c, according to theprescriptions of mathematical statistics, c is situated in such-and-suchconfidence interval with confidence probability P = 0.99 and within amore narrow interval with P = 0.95.I also refer to physicists in the sequel. It was in physics that thebasis of modern exact natural science had been laid, the largest amountof experience of complicated and subtle experimentation accumulatedand a developed culture of a sound treatment of experimental data hadbeen achieved. On the other hand, it was physics that provided theexample of applying mathematical structures which is now oftenrecognized not favourably enough for other fundamental and applieddisciplines. I return to that problem at the end of my booklet.Its main aim is to describe the principles of such a treatment of datathat abstains from mentioning confidence probabilities. Theseprinciples had appeared even before mathematical statistics had;indeed, appeared at the same time as the first quantitative experimentalresults in natural science did. However, they were reflected in thetheory of probability only much later during the process of thedevelopment of the approach connected with Mises. This approach hasbeen vividly discussed for decades, see my papers and textbooks(1976, 1977, 1987b; 1978a; 1979).The connection of that Mises approach with the principles andmethods different from those of mathematical statistics is fundamentaland the contents of this booklet is therefore largely reduced to aconsistent although only understandably sketchy description of thatapproach. Such an exposition is still lacking in the literature easilyread by a broad circle of readers.I am concentrating on the problems of interpretation and practicalapplication of stochastic notions. Unlike the solution of purelymathematical issues, any answers to such problems are always to alarge extent arguable and the reader ought to take it into account. I amdescribing an approach noticeably different from that of the standardtreatises and most works on probability theory and mathematicalstatistics and I repeat that my point of view is not at all new. Itsextreme version is nicely expressed, for example, by Anscombe [1967,p. 3 note]: it is inadmissible to identify statistics with the grotesquephenomenon generally known as mathematical statistics.1. Introductory Remarks about ForecastingThe final aim of research in both fundamental and applied naturalscience is a reliable forecast of the results of future experiments. Byexperiment I mean not only investigative, reconnaissance trials, but123
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 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 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
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: Kolman E. (1939 Russian), Perversio
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?