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Biometrika 1969 56(3):641-650; doi:10.1093/biomet/56.3.641
© 1969 by Biometrika Trust
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Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample

G. P. SILLITTO

Malvern Link, Worcestershire

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.


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