Representation of one orthonormal set in terms of another

Abstract

SUMMARY Different methods of constructing orthonormal simple functions on a multinomial distribution are considered, and the functions so constructed are compared with the orthonormal polynomials associated with the underlying continuous distribution. The particular cases investigated are those in which the underlying distribution is either normal or rectangular. Step functions, which are orthonormal on a continuous distribution, will be called orthonormal simple functions to distinguish them from the orthonormal polynomials on that distribution. Some orthonormal simple functions are 'closer' to the orthonormal polynomials than others, and a measure of 'closeness' is the representation of one orthonormal set in terms of another, as defined in ? 2- 1. Good representation of one orthonormal set in terms of another has been found useful before. Hamdan (1962, 1964) gives alternatives to Neyman's (1937) criterion for the 'smooth' test of goodness of fit. His criteria are simpler to compute than Neyman's and are almost as powerful. They are defined by replacing the Legendre polynomials by a set of orthonormal simple functions, provided the rectangular distribution is well represented by the simple functions. Lancaster & Hamdan (1964) note that Pearson's (1913) method of estimating the correlation coefficient in a contingency table is not good if the first Hermite polynomial is not well represented in the Helmert orthonormal sets. The numerical results of Tables 1 and 2 show that for not unduly coarse partitions of the interval (- o, oo) for the normal distribution and (0, 1) for the rectangular distribution, the Hermite and Legendre polynomials are well represented by the orthonormal simple functions.