Two classes of multiplicative algorithms for constructing optimizing distributions

• Published in: Computational statistics & data analysis Vol. 51; no. 3; pp. 1591 - 1601
• Main Authors: Torsney, B., Mandal, S.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.12.2006; Elsevier Science; Elsevier
• Series: Computational Statistics & Data Analysis
• Subjects: Distribution theory; Exact sciences and technology; General topics; Mathematics; Multiplicative algorithms; Multivariate analysis; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Optimal designs; Optimal distributions on spaces; Optimality conditions; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Vertex directional derivatives; Optimality conditions; Multiplicative algorithms; Vertex directional derivatives; Optimal distributions on spaces; Optimal designs; Partial derivative; Data analysis; Optimal designs: Vertex directional derivatives; Algorithm; Convergence; Statistical computation; Experimental design; Distribution function; Function derivative; Summation
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/j.csda.2006.05.014

We construct approximate optimizing distributions p j by maximizing a criterion function subject to the basic constraints on p j of nonnegativity and summation to unity. We use a class of multiplicative algorithms, indexed by a function f ( . ) , which depends on the derivatives of the criterion function. The function may depend on one or more free parameters. To improve the convergence, we consider two approaches—Approach I and Approach II. Approach II is reported in Torsney and Mandal [2004. Multiplicative algorithms for constructing optimizing distributions: further developments. In: Bucchianico, A.D., Läuter, H., Wynn, H.P. (Ed.), MODA 7—Advances in Model-Oriented Design and Analysis. Physica, Heidelberg, pp. 163–171]. In Approach I, the criterion function can have only positive partial derivatives, whereas in Approach II the criterion can have both positive and negative derivatives. We consider objective choices of the class of functions f ( . ) for the two approaches. Considerable improvements in convergence are seen for each of the approaches.