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

More documents

Recommendations

Info

sigma in absolute value and is highly significant. The first series ofexperiments is not, strictly speaking, a confirmation of the Mendelianlaws.Let us ask ourselves now, how large should the deviation be fromthose laws that we ought to admit when considering this series ofexperiments. It is certainly possible to say at once now that thediscussion is pointless when declaring that such a result compels us todoubt the presence of a statistical ensemble; or, roughly the same, todoubt the independence of the separate outcomes of the experiments.But let us try to manage by less cruel means. Suppose that eachplant reveals the recessive indication independently from others, butthat the probability of success (appearance of a plant with potato-likeleaves) p differs from 1/4: p = 1/4 + ∆p. How large should ∆p be forexplaining the observed shift of the empirical distribution function?Suppose that ∆p = − 1/40. We thought that the magnitude (2.6) with p= p 0 = 3/4 and q = q 0 = 3/4 has a standard normal distribution;actually, this will be true for[µ − n( p + ∆q)]0µ* = .n( p + ∆ p)( q + ∆q)0 0When calculating the difference betweenneglect the change of the denominator so that* * n∆pµ0≈ µ + .np q0 0*µ0and*µ , we mayThe magnitudes n differ in different experiments, but, according toTable 1, np = np 0 = n/4 mostly exceeds 50, so that n ≥ 200. Therefore,the systematic shift isn ∆ p 4= n ∆ p ≈ 30∆pnp q 30 0and ∆p = − 1/40 quite well explains the systematic shift of − 0.64.An estimate by naked eye using the dotted line on Fig. 3 provides− 0.58, little differing from − 0.64 since the mean square deviation ofthe arithmetic mean is 1/ 11 ≈ 0.30.At present, there are tables of the distribution of the statisticλ′ = sup |F(x) − F n (x)| (2.7)xalso for finite values of n, see for example Bolshev & Smirnov (1967).For the first series of observations (n = 11) that statistic is 0.28. It isvery moderately significant for levels higher than 20%.Consequently, when applying this test, we are not compelled toconsider that the data of the first series reject the applicability of the104
Mendelian laws. It is not sufficiently clear which conclusion has morenatural scientific sense: either that the data do not agree with thoselaws, but that the discrepancy can be understood by a slight change ofp (equal to 10%); or, that somewhat reluctantly we may suppose thatthere is no obvious contradiction with those laws.However, Enin provides some explanation of the possiblediscrepancy: the plants in the hothouse sown in February suffered froma shortage of heat and light and a considerable part of the sproutedseeds perished. Plants having a recessive indication could have wellhad a somewhat lower probability of survival (which should bechecked by a special experiment). The final results of the first seriescan be considered as some modest confirmation of the Mendelianlaws.We turn now to the second series. The pertinent empiricaldistribution function on Fig. 3 is only badly smoothed by a straightline (according, however, to my somewhat subjective opinion). In anycase, the scatter of the observations is essentially less than supposed bythe standard normal distribution. The most simple way to show it by astatistical criterion is to calculate the sum of the squares of theobservations. It is equal to 2.85 whereas its distribution (if the checkedhypothesis is valid) is the chi-squared law with 14 degrees of freedom.As indicated by the tables of that law, that value is thus practicallyimpossible. The value of the statistics (2.7) is 0.33; with n = 14 that issignificant at about the 5% level.The shift of the first series of observations was in some wayreasonably explained; the second series has an insignificant shift (thesample mean is − 0.21) but an essentially smaller than supposedvariance. The Mendelian laws are thus obeyed more precisely thansupposed which is hardly possible. The most probable statisticalconclusion is that the results were tampered with deliberately or not.The corruption of normality of the distribution (the impossibility ofsmoothing the empirical distribution by a straight line) also indicatessome defect; however, for the given number of observations thisconclusion would be difficult to justify by a statistical test.In general, as far as was possible to ascertain, the trouble isapparently that the experimental data are not provided in full. And so,it is possible to confirm the Mendelian laws while intending to refutethem, and it is also possible to throw them into doubt when intendingto confirm them, and all of this is revealed by a purely statisticalinvestigation.Here, we encountered a curious violation of the order beingestablished in mathematical statistics. When acting strictlyscientifically, statistical tests should be chosen beforehand and theexperiment carried out and the verdict passed only afterwards.Actually, the tests are more often chosen by issuing from peculiaritiesof the material noted by naked eye. They serve for checking whetherthese peculiarities are statistically significant or not. However, havingestablished in our case that useful are tests based on the sample meanand the sum of the squares of the sample values, we could have, whenanalyzing new similar material, strictly followed statistical science.105
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: *several dozen. The totality µ ica
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?