Robust estimation of variance coefficient with excess of observation errors distribution (VR_(gamma)-estimation). Part I. theory

摘要

A method of estimation which allows to obtain robust, efficient, invariant and unbiased estimator of the variance coefficient sigma_0~2 is derived in the work. The coefficient sigma_0~2 occurs in the model of covariance matrix of observation resultsSIGMA = sigma_0~2Q (Q - known cofactor matrix). It is assumed that the class of square estimators of the following general form sigma_0~2 = epsilon~T OMEGA epsilon, where OMEGA is unknown, symmetric matrix (epsilon - vector of observation errors). The method presended in the paper is the development of proposal described by the author in another paper (wisniewski 1999). It consists in searching for such a matrix OMEGA R_(gamma) which, on one hand, guarantees invariance and unbiasness of the estimator,and on the other hand, minimises variance ofthe quadratic form epsilon~T(R_c(epsilon)*OMEGA_(R_(gamma)))epsilon, where R_c(epsilon) is a certain function representing gross errors (* - Hadamard product). The estimator obtained in such a way, is the mosteffective for given matrices R_c = R_c(epsilon) and gamma (matrix of observation error excess coefficients). The first part of the paper presents the theory of the proposed method of estimation (called VR_(gamma)-estimation), where as the, fundamental content of the second part of the paper are empirical analyses of the method, including proposals concerning the creation of the functions R_c(epsilon).