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

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