7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Smirnov, Romanovsky and others made a number of interesting investigations concerningthe estimation of statistical hypotheses. In §4 devoted to distribution-free methods we shalldescribe some studies of the tests of goodness-of-fit and homogeneity.3. The variational series is known to be the ordered sequence 1 (n) < 2 (n) < … < n (n) ofobservations of some random variable with continuous distribution F(x). Many scientistsexamined the regularities obeyed by the terms of a variational series. Already in 1935 and1937 Smirnov [3; 7] systematically studied its central terms. Gnedenko’s examination [17;25] of the limiting distributions for the maximal terms appeared somewhat later. Thesecontributions served as points of departure for further research.Smirnov [15] devoted a considerable study to the limiting distributions both for the centraland the extreme terms with a constant rank number. A sequence of terms k (n) of a variationalseries is called central if k/n , 0 < < 1, as n . Magnitudes k (n) are called extremeterms of a variational series if either the subscript k, or the difference (n – k) are constant.Denote the distribution of k (n) by nk (x) and call (x) the limiting distribution of k (n) (k =Const), if, after adequately choosing the constants a n and b n , nk (a n x + b n ) (x) as n .The following Smirnov theorem describes the class of the limiting distributions. The properlimiting distributions for a sequence with a constant number k can only be of three types: (k) x(x) = [1/(k – 1)!] α (k) | x| (x) = [1/(k – 1)!] −α (k) e(x) = [1/(k – 1)!] x000e –x x k–1 dx, x, > 0;e –x x k–1 dx, x < 0, > 0; (1)e –x x k–1 dx, – < x < .Gnedenko earlier derived the limiting distributions for the minimal term; they can certainlybe obtained from (1) by taking k = 1. The conditions for attraction to each of these threelimiting distributions exactly coincide with those determined by Gnedenko [25] for the caseof the minimal (maximal) term.Smirnov established a number of interesting regularities for the central terms. We shallspeak about a normal -attraction if there exists such a distribution (x) that, if[(k/n) – ]n 0 as n and the constants a n and b n (which generally depend on ) are adequately chosen,P{[( k (n) – b n )/a n ] < x} (x) as n .The following four types exhaust the class of the limiting distributions having domains ofnormal -attraction: (1) (x) = (1/ (2) (x) = (1/αcx2 π ) −∞2 π ) − −∞αc|x|exp(– x 2 /2)dx, x ≥ 0, c, > 0; (1) (x) = 0 if y < 0;exp(– x 2 /2)dx, x < 0; (2) (x) = 1 if x, c, > 0;