Estimating a Polya Frequency Function

The problem of estimating a unimodal density and its mode has attracted a wide interest in the literature, beginning with the work of Barlow [1], Prakasa Rao, [7], Robertson, [8], and Wegman [13], [14] and continuing through [9], [2], [3], [5], and [4], who can be consulted for further references. Asymptotic properties of the maximum likelihood estimators have been developed but may be messy and suffer from some inconsistency in the region near the mode. Kernel estimation can avoid the inconsistency, but must confront the choice of a bandwidth. Here we investigate a smaller, easier version of the problem, estimating a Polya frequency function. By a Polya frequency function, we mean a density f whose logarithm is concave over the support of f . Equivalently, a function f whose logarithm is concave in the sense of Rockafellar [10]. Such functions are automatically unimodal. Moreover, an estimated Polya frequency function supplies its own estimate of the mode. There is no need to estimate the mode seperately. The non-parametric maximum likelihood estimator [hereafter, NPMLE] for this problem is derived in Section 2 and shown to be consistent in Section 3. Simulations are reported in Section 4. ∗Supported by Horace Rackham Graduate School and National Science Foundation †Supported by National Science Foundation AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35