Semi-infinite programming duality for order restricted statistical inference models

Abstract

The equivalence of multinomial maximum likelihood and the isotonic projection problem:

$$\inf \left\{ {\sum\limits_{i = 1}^n {p_i \ln (p_i /r_i )|p \in K,a convex} {\mathbf{ }}subset of the probability vectors{\mathbf{ }}p} \right\}$$

can be established using Fenchel's Duality Theorem and subgradient and complementary slackness relationships of convex analysis, all taking place over the real numbers.

In this paper non-Archimedean polynomial subgradients (Jeroslow/Kortanek '71, Blair '74, Borwein '80, and Kortanek/Soyster '81) are employed for the case where some of the observed values of the random vector are zero, corresponding to “zero counts in the traditional multinomial setting.” With an appropriate linear semi-infinite programming dual pair it is shown that a vector solves the multinomial problem if and only if it converts to a solution of the isotonic projection problem. The development parallels the one of Robertson/Wright/Dykstra '88, where for the zero counts case the authors adjoin “-∞” to the real numbers and define ln(0)=-∞.