Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample

Abstract

Approximations to the mathematical relation between the variate value z and the unknown distribution function F of the continuous, univariate population from which a sample is available are obtained in which z is expressed as a series of polynomials in F ; the coefficients are the expectations of linear combinations of the order statistics of the sample. A particular system of orthogonal polynomials and a particular system of n linearly independent linear systematic statistics of a sample of n emerge naturally as apt for the purpose. General relations are found for the variances and covariances of these statistics, thus enabling them to be used as a vector basis in terms of which other linear combinations of the order statistics can be expressed and their variances and mutual covariances investigated.