Some generalizations of method of moments in probability density estimation by orthogonal series

A problem of probability density function estimation for continuous random variable is considered in the paper. Actuality of this problem for science and technology including aerospace branch is discussed. Building of the probability density functions estimation by orthonormal series is considered. Applying of the method of moments to statistical estimating series coefficients is investigated, and some generalizations are suggested. Within the scope of method of moments the author uses consecutive ordinary moments of random variable. In this case the calculation of series coefficients estimations is reduced to solving system of linear equations. The paper has the proof of the theorem that in specified conditions of orthonormal system choice, suggested generalization of the method of moments includes the ordinary method of moments and also more widespread estimating coefficients method by sample means of orthonormal functions as particular cases. The paper shows that in the case of using Legendre’s orthonormal system the considered methods are identical. The properties of built estimations are researched, mathematical expectations and covariance matrix are founded. The paper shows that in particular cases the estimations of series coefficients are biased. The estimations of series coefficients, which are calculated by suggested method, were used for building estimation of probability density function. The paper has results of numerical calculations of quality functional values for built estimations of probability density function. The calculations are carried out for uniformly distributed random value. The author compares the suggested method with the widespread estimating method with using Chebyshev’s and trigonometric orthonormal systems. Research shows that generalized method of moments gives substantially better result on small samples.