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

values of (2.1) realized in the n experiments and is therefore itselfrandom. In addition, there exists a non-random (theoretical)distribution functionF(x) = P[ξ < x] = P[x i < x] (2.3)of each result of the experiment.Kolmogorov proved that at n → ∞ the magnitudeλ = sup n | F( x) − F ( x) |(2.4)nhas some standard distribution (the Kolmogorov distribution); thesupremum is taken over the values of x. This result is valid under asingle assumption that F(x) is continuous. Now not only theasymptotic distribution of (2.4) is known, but also its distributions at n= 2, 3, ...The practical sense of the empirical distribution function F n (x)consists, first of all, in that its graph vividly represents the samplevalues (2.1). In a certain sense this function at sufficiently large valuesof n resembles the theoretical distribution function F(x). [...]There also exists another method of representation of a samplecalled histogram [...] Given a large number of observations, itresembles the density of distribution of random variable ξ. However, itis only expressive (and almost independent from the choice of theintervals of grouping) for the number of observations of the order of atleast a few tens. The histogram is more commonly used, but in allcases I decidedly prefer to apply the empirical distribution function.The Kolmogorov criterion based on statistics λ, see (2.4), can beapplied for testing the fit of the supposed theoretical law F(x) to theobservational data (2.1) represented by function (2.2). However, thattheoretical law ought to be precisely known. A common (but graduallybeing abandoned) mistake was the application of the Kolmogorovcriterion for testing the hypothesis of the kind The theoreticaldistribution function is normal. Indeed, the normal law is onlydetermined to the choice of its parameters a (the mean) and σ (meansquare scatter). In the hypothesis formulated just above theseparameters are not mentioned; it is assumed that they are determinedby sample data, naturally through the estimators1x s x xn2 2; = ∑ (i− ) .n −1i=1Thus, instead of statistic (2.4), the statisticx − xsup n | F0[ ] − Fn( x) |(2.5)sis meant. Here, F 0 is the standard normal law N(0, 1).Statistic (2.5) differs from (2.4) in that instead of F(x) it includes F 0which depends on (2.1), x and s and is therefore random. Typical101