7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Exactly in such a way were determined the various confidence limits estimating theparameters by sampling and the criteria for testing hypotheses now constituting the mainstatistical tool described in all pertinent courses.In our time, when statistical methods are being applied under conditions very unlike innature one to another, the assumptions of the classical parametric statistics are unable tocover all the field of issues that we encounter. In practice, examining the distributions ofrandom variables, we ought in many cases to restrict the problem only by very generalsuppositions (only assuming, for example, continuity, differentiability, etc). Tests orconfidence estimates determined by issuing from these general premises were indeeddesignated non-parametric which stressed their distinction from their counterparts in classicalstatistics.Practitioners have been applying some non-parametric methods for a long time. Thus, itwas known how to obtain confidence limits for the theoretical quantiles of an unknowndistribution function (under the sole assumption of continuity) given the terms of thevariational series; and, in particular, how to estimate the position of the theoretical median.The application of the coefficients of rank correlation and of various tests of randomnessbased on the theory of runs were also known long ago.Already during the 1930s – 1940s Soviet mathematicians achieved considerably deeperfindings in the area of non-parametric statistics. Here, we only mention the remarkable test ofthe agreement between an empirical function of distribution F n (x) and the hypotheticallyadmitted theoretical law F(x). The appropriate theorem provides an asymptotic distribution ofthe criterionD n = sup|F n (x) – F(x)|whose complete theory is based on a theorem due to Kolmogorov. Only continuity of F(x) ishere demanded and D obeys a universal law distribution independent of the type of F(x).Similar tests independent of the type of the theoretical distribution function were laterobtained in various forms and for various cases of testing hypotheses.[5] The new direction attracted the attention of many eminent mathematicians and is nowone of the most productive for the general development of statistical science. Theindependence from the kind of distribution enables to apply much more justifiably nonparametrictests in the most various situations. Such tests also possess a property veryimportant for applications: they allow the treatment of data admitting either no quantitativeexpression at all (although capable of being ordered by magnitude) or only a quantitativeestimate on a nominal scale. And the calculations demanded here are considerably simpler.True, the transition to non-parametric methods, especially for small samples, is connectedwith a rather essential loss of information and the efficiency of the new methods as comparedwith the classical methodology is sometimes low. However, the latest investigations (Pitman,Lehmann, Z. Birnbaum, Wolfowitz, van der Waerden, Smirnov, Chibisov and many others)show that there exist non-parametric tests which are hardly inferior in this respect to, andsometimes even better than optimal tests for certain alternatives.Comparative efficiency is understood here as the ratio of the sample sizes for which thecompared tests possess equal power for given alternatives (assuming of course that theirsignificance levels are also equal). Thus, the Wilcoxon test concerning the shift of thelocation parameter under normality has a limiting efficiency of e = 3/ 0.95 and is thushardly inferior to the well-known Student criterion. And it was shown that under the sameconditions the sequential non-parametric sign tests possesses a considerable advantage overthe Student criterion: its efficiency is 1.3. A special investigation revealed the followingimportant circumstance: supposing that the well-known classical 2 test demands n