7. Probability and Statistics Soviet Essays - Sheynin, Oscar

ehavior of the law of distribution of the empirical correlation coefficient (derived byFisher). Smirnov [10] discovered the distribution of the maximal deviation (normed by theempirical variance) of observations from their empirical mean. This enabled him to makemore precise the well-known Chauvenet rule for rejecting outliers 1 .Smirnov [2; 6] studied the terms of the variational series, i.e., of the observed values of arandom variable arranged in order of ascending magnitude, and established the appropriatelimit laws under rather general assumptions. Gnedenko [15; 23] obtained interesting resultsabout the distribution of the extreme terms of such series. There are three and only threelimiting distributions of these terms (Fisher & Tippett and Mises); for the maximal term theseare (x) = 0 if x ≤ 0 and = exp (– x – ) otherwise; (x) = exp [– (– x) ] if x ≤ 0 and = 1 otherwise, > 0;(x) = exp (– e –x ), |x| < + .By very subtle methods Gnedenko ascertained necessary and sufficient conditions for theoccurrence of each of these and delimited in full the domains of their attraction. The works ofGumbel show that this theory finds applications in hydrological calculations (volumes ofreservoirs), investigations of extreme age brackets, civil engineering, etc. Making use ofGnedenko’s method, Smirnov, in a not yet published paper, presents an exhaustive to acertain extent classification of the limit laws for the central terms of the variational series andof the domains of their attraction.Problems connected with a rational construction of statistics most effectively estimatingthe parameters of a theoretical law of distribution for a given size of the sample are urgent formodern science. Here, the classical approach, when the estimated parameter is considered asa random variable with some prior distribution is in most cases fruitless and the veryassumption that a prior distribution exits is often unjustified. Fisher and Neyman put forwarda new broad concept. It sees the main problem of the statistical method in establishingsubstantiated rules aiming at selecting hypotheses compatible with the observed data fromamong those admissible in the given concrete area of research. These rules should, first, besufficiently reliable, so that, when used regularly, they would practically seldom lead tomistaken results; and, second, they should be the most effective, so that, after accounting forthe observational data, their use would narrow the set of admissible hypotheses as much aspossible. The measure of the good quality of a statistical rule is the confidence coeffcientdefined as the lower bound of the probabilities of a correct conclusion resulting from the rule.Fisher and Neyman developed methods that allow, when only issuing from the sample data(without introducing prior probabilities), to indicate confidence boundaries that correspondto the assumed confidence coefficient and cover the estimated parameter of the generalpopulation. The revision of the already established methodology and the development of newideas is the main channel of modern scientific work for those engaged in this domain.Kolmogorov [43] presented an original interpretation of these ideas which specifies somesubtle logical points as applied to the simplest problem of estimating the parameters of theGauss law by a restricted number of observations. Bernstein [37] indicated the difficultiesconnected with Fisher’s concept which restrict the applicability of his methods by conditionsjustified within the boundaries of the classical theorems. In the final analysis, the estimationof the efficiency of statistical rules is inseparable from an accurate notion of the aim of thestatistical inmvestigation. The peculiarity of the logical situation and the uncommonness ofthe introduced concepts led to a number of mistaken interpretations (Fisher himself was alsoguilty here), but the fruitfulness of the new way is obvious. For our science, the furtherdevelopment of the appropriate problems is therefore an urgent necessity. Romanovky [44]and Brodovitsky [2] described and worked out a number of pertinent problems. Again