7. Probability and Statistics Soviet Essays - Sheynin, Oscar

conditional variance and determined various correlation invariants. For the normallydistributed n-dimensional random vector the normal law is completely defined by the vectorof the expectation and the tensor of variance. Obukhov’s proposition on the canonicalexpansion of the correlation density that he proved reduces the study of multidimensionalvectorial correlations to the case of one-dimensional vectors.Romanovsky [21] studied various problems concerned with the connection of qualitativeindications (the theory of association).When determining the number and the size of the samples needed for establishing themean value of some quantitative indication distributed over a certain area (for example, ofthe harvest, or the content of metal in ore), we have to allow for correlation between differentpoints of the area. Boiarsky [3] made an interesting attempt to isolate continuous isotropicrandom fields of the Markov type and Obukhov [5] obtained more general results.Romanovsky [21], Nemchinov [1], Mitropolsky [2; 3; 4] and Lagunov [1] studied problemsconnected with the determination of the equations of regression.2. Distribution of Sample Statistics. Estimating the Parameters of theLaws of DistributionThe problems of sampling, that is, of the methods of approximately determining variouscharacteristics of the general population given the empirical material, can be very diversedepending on the nature of the theoretical law of distribution and the organization of theobservations. A rational choice of such functions of the observations (the choice of statistics,as Fisher called them) which provide, under given conditions, the best (in a certain sense)approximation of (information on) the estimated theoretical magnitudes (for example, of/onthe parameters of the law of distribution) is a complicated problem. The precision of theapproximation can be estimated in full if the law of distribution of the sample statistic isknown. In this case, it is also possible to evaluate the greater or lesser suitability of thechosen statistic as compared with other possible functions to serve as an approximatemeasure of the estimated parameter. The investigation of the laws of distribution of thevarious kinds of empirical means (means, variances, correlation coefficients, etc) is thereforeone of the most important problems of mathematical statistics. Those mostly studied,naturally occurred to be samples from normal populations, and many contributions ofEnglish and American statisticians headed by Fisher were devoted to this subject. Thepossibility of entirely describing the normal distribution by a small number of parameters andthe comparative simplicity of calculations clear the way for a deep analysis of the variousrelations between the general population and the sample that represents it.Prominent and universally recognized achievements in this domain are due toRomanovsky. Issuing from the notions of the British school, his writings [8; 10; 11; 13; 18;19; 21] nevertheless advantageously differ since they are based on rigorous methodologicallines and are free from a rather considerable jumble of the main assumptions; indeed, heovercame the confusion of empirical and stochastic elements so characteristic of the Englishstatisticians. With considerable analytic mastery Romanovsky applies the method ofgenerating functions which leads to peculiar inversion problems in the theory of integralequations of the first kind.He was the first to derive rigorously the laws of distribution of the well-known Student –Fisher t- and z-criteria, of empirical coefficients of regression and of a number of otherstatistics. A summary of his main results can be found in his well-known treatise [37] thatplayed a fundamental part in the heightening of the mathematical level of the statisticalthought.Kuznetsov [2] studied the distributions of the length and the argument of a radius vectorby the normal distribution of its components. Kuzmin [2] investigated the asymptotic