Conclusions

Abstract

Thus in mathematical statistics an ‘essentially multivariate’ approach to multi-dimensional problems is developed, which considers specific effects produced by estimating an essentially large number of parameters (that is, a number of parameters comparable in magnitude with the sample size). It differs by an account of phenomena produced by the estimation of a large number of parameters over samples of a limited size. We can say that a new branch of mathematical statistics is developed which may be called a theory of essentially multivariate, or, more precisely, of essentially multi-parametric problems. In this theory a principal role is played by the ratio of the observation dimension n to sample size N. With a finite n/N we must take into account the systematic accumulation of errors of a large number of estimators and ari additional averaging (‘self-averaging’, ‘mixing’) over a great number of weakly dependent variables. For n comparable with N, functions uniformly depending on the large number of variables display a finite bias, additional multiples and other effects of the order of magnitude n/N, along with the decrease of variances as 1/N if n/N is bounded. These ‘essentially multivariate’ effects prove to be of a special importance in problems related to the inversion of sample covariance matrices. Here, in simple cases, a factor N/(N — n — 2) appears as a result of ill-conditioned solution of the inversion problem. For n comparable with N,standard linear solutions prove to be unstable or may not exist making it impossible to treat data of high dimension.