International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 199-200
Section 2.1.7.2. Mathematical background^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England |
Suppose that is a p.d.f. which accurately describes the experimental distribution of the random variable x, where x is related to a sum of random variables and can be assumed to obey (to some approximation) an ideal p.d.f., say , based on the central-limit theorem. In the correction-factor approach we seek to represent as where are coefficients which depend on the cause of the deviation of from the central-limit theorem approximation and are suitably chosen functions of x. A choice of the set is deemed suitable, if only from a practical point of view, if it allows the convenient introduction of the cause of the above deviation of into the expansion coefficients . This requirement is satisfied – also from a theoretical point of view – by taking as a set of polynomials which are orthogonal with respect to the ideal p.d.f., taken as their weight function (e.g. Cramér, 1951). That is, the functions so chosen have to obey the relationship where is the range of existence of all the functions involved. It can be readily shown that the coefficients are given by where the brackets in equation (2.1.7.3) denote averaging with respect to the unknown p.d.f. and is the coefficient of the nth power of x in the polynomial . The coefficients are thus directly related to the moments of the non-ideal distribution and the coefficients of the powers of x in the orthogonal polynomials. The latter coefficients can be obtained by the Gram–Schmidt procedure (e.g. Spiegel, 1974), or by direct use of the Szegö determinants (e.g. Cramér, 1951), for any weight function that has finite moments. However, the feasibility of the present approach depends on our ability to obtain the moments without the knowledge of the non-ideal p.d.f., .
References
Cramér, H. (1951). Mathematical methods of statistics. Princeton University Press.Google ScholarSpiegel, M. R. (1974). Theory and problems of Fourier analysis. Schaum's Outline Series. New York: McGraw-Hill.Google Scholar