Application aspects of G-analysis

Recent developments of the general approach to multivariate analysis (G-analysis), where the dimension of variables Π is large and comparable with the sample size JV, are discussed. Its platform is a non-linear functional relation between the principle parts of spectral functions of true and sample covariance matrices. The remainder terms are small if all four moments of variables are bounded and a special average measure of quadratic forms is small. Three statistical problems are considered under the G-analysis approach (for normal populations): (1) estimating the mean of a random vector with independent components and variance 1, (2) discrimination of two populations with common unknown covariance matrix, and (3) linear regression in case of random predictors and random response. Regularized and generalized procedures are considered depending on an arbitrary a priori function of finite variation. Under the asymptotics η —> OO, N —l· OO, U l'N —> y > 0, limit formulas are derived for quality functionals of these procedures. The extremum problems are partially solved.