## The Annals of Statistics

### An iterative procedure for general probability measures to obtain I-projections onto intersections of convex sets

#### Abstract

The iterative proportional fitting procedure (IPFP) was introduced formally by Deming and Stephan in 1940. For bivariate densities, this procedure has been investigated by Kullback and Rüschendorf. 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.

#### Article information

Source
Ann. Statist., Volume 34, Number 2 (2006), 878-902.

Dates
First available in Project Euclid: 27 June 2006

https://projecteuclid.org/euclid.aos/1151418244

Digital Object Identifier
doi:10.1214/009053606000000056

Mathematical Reviews number (MathSciNet)
MR2283396

Zentralblatt MATH identifier
1117.62003

#### Citation

Bhattacharya, Bhaskar. An iterative procedure for general probability measures to obtain I-projections onto intersections of convex sets. Ann. Statist. 34 (2006), no. 2, 878--902. doi:10.1214/009053606000000056. https://projecteuclid.org/euclid.aos/1151418244

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