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An iterative procedure for general probability measures to obtain $$I$$-projections onto intersections of convex sets. (English) Zbl 1117.62003
Summary: The iterative proportional fitting procedure (IPFP) was introduced formally by W. E. Deming and F. F. Stephan [Ann. Math. Stat., Ann. Arbor 11, 427–444 (1940; JFM 66.0652.02)]. For bivariate densities, this procedure has been investigated by S. Kullback [Ann. Math. Stat. 39, 1236–1243 (1968; Zbl 0165.20303)] and L. Rüschendorf [Ann. Stat. 23, No. 4, 1160–1174 (1995; Zbl 0851.62038)]. It is well known that the IPFP is a sequence of successive $$I$$-projections onto sets of probability measures with fixed marginals. However, when finding the $$I$$-projection onto the intersection of arbitrary closed, convex sets (e.g., marginal stochastic orders), a sequence of successive $$I$$-projections onto these sets may not lead to the actual solution. Addressing this situation, we present a new iterative $$I$$-projection algorithm. Under reasonable assumptions and using tools from Fenchel duality, convergence of this algorithm to the true solution is shown. The cases of infinite dimensional IPFP and marginal stochastic orders are worked out in this context.

MSC:
 62A01 Foundations and philosophical topics in statistics 60B10 Convergence of probability measures 65K10 Numerical optimization and variational techniques 60E15 Inequalities; stochastic orderings 62B10 Statistical aspects of information-theoretic topics 62B99 Sufficiency and information 62H99 Multivariate analysis
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