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

Another group of tests is connected with the choice of the degreeand the number of terms of the approximating polynomial. Here, wealso deal with v i (2.1) and test what happens when they areapproximated by various polynomials up to the third degree inclusive.It is obvious that the polynomial sought includes a free term. Then weadd, in turn, terms of the first, second and third degree. The bestimprovement of approximation is reached when polynomials of thetypec 0 + c 2 t 2 (2.2)are chosen.And now we check that the addition of terms of the first and thirddegree to it does not significantly improve the approximation; fordetails, see Belova et al (1965, 1967). After all these checks webecome sure that applying a polynomial (2.2) we have indeed ascompletely as was possible elicited the determinate component fromthe available data.However, having happily concluded the tests of the hypothesesconnected with the smoothing, we do not at all check the mainhypothesis, that the probability of the failure of a unit area ofinsulation does not depend on the constructive or operationalpeculiarities of the pertinent machine. Indeed, we are only checkingwhether the magnitudes µ i are obeying the Poisson distribution (and, inpart, whether they are independent).However, that distribution also occurs when the probabilities offailures occurring on different areas of the insulation are unequal(provided all the probabilities are sufficiently low). The mostimportant hypothesis of statistical homogeneity of the various unitareas of insulation is yet left unchecked and can not be checked byissuing from the generalized data of Fig. 1 4 . At the same time mostinteresting is exactly the isolation and study of machines with high andlow break-down rates (or a confirmation that all of them have the samerate of failures). We will see now how these problems can be solved.2.3. Check of statistical homogeneity. The most importantcondition of acquiring a statistically homogeneous totality, or, so tosay, the most important mystery of the statistical art consists incarefully selecting the material to be studied. Thus, the data of Fig. 1does not include failures of the insulation occurring because of causes[of various causes of its random damage]. We supposed that suchcauses, although usually called random, are not random in thestochastic sense since they are not statistically stable.The selection of material was made easier by the fact that a failureof a large machine is an extreme event whose causes are thoroughlyinvestigated and duly registered. The most suitable for including afailure into statistical treatment was the formulation local defect ofinsulation. In general, however, all failures were included if an aliencause was not clearly indicated. Failures included into statisticaltreatment composed about a half of all the failures of insulation.When selecting material, the statistician must invariably keep to someprinciple once and for all.60