7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Romanovsky [45] and Kolmogorov [46] revised the canonical explication of the method ofleast squares and the theory of errors in the spirit of the new ideas 2 .3. Statistical Estimation of the Laws of Distribution. Testing theGoodness of Fit. Hypothesis TestingAs stated in the Introduction, one of the main problems of mathematical statistics is toreconstruct the theoretical law of distribution of the general population, given the empiricaldistribution of the sample. For a one-dimensional population this problem is reduced toestablishing the distribution function F(x) of some random variable by issuing fromindependent observations. If the variable is discrete and takes s values x 1 , x 2 , …, x s the dataare separated in a natural way into s groups of m 1 , m 2 , …, m s observa-tions (m 1 + m 2 + …+m s = n) corresponding to each of the possible values of . The probable measure ofapproximation attained when the size of the sample is not too small can be estimated byasymptotic formulas; inversely, they also enable us to determine the adequate number ofobservations for guaranteeing, with a given degree of probability, the necessary closeness ofthe empirical and the theoretical distributions of probability. If the theoretical distribution isassumed to be known, and the admissibility of the observed deviation is questioned, theanswer is rather fully obtained by means of the well-known Pearsonian chi-squared testwhose rigorous theory was presented by Romanovsky [22]. For a continuous distributionfunction F(x) all such problems become considerably more difficult; only recently theMoscow mathematicians Glivenko, Kolmogorov and Smirnov rigorously solved them.The relative frequency S n (x) of those observations that do not exceed the given number x iscalled the empirical distribution function; it is depicted by a step-curve. Glivenko [4] offeredthe first rigorous proof that the empirical curve uniformly converges to the continuoustheoretical law with probability 1. This gifted mathematician, who died prematurely, basedhis reasoning on a very abstract notion of the law of large numbers in functional spaces anddeveloped it in a number of interesting writings. He [4] discovered the following simpleestimate (from which the abovementioned theorem was immediately derived):LetD n = sup |S n (x) – F(x)|, |x| < + ,then, for any continuous F(x) and < 0,P(D n < ) > 1 – [1/( 6 n 2 )].Kolmogorov [16] proved a more precise proposition:Let n () = P(D n ≤ /n), > 0 > 0.Then, for any continuous law of distribution F(x), the sequence of functions n () tends tothe limiting law() = ∞−∞ n=(– 1) k exp(– 2k 2 2 )as n increases.