Estimating a Polya Frequency Function
Estimating a Polya Frequency Function
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估计 Polya 频率函数
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Mary Meyer
中科院分区：
文献类型：
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作者：
J. K. Pal;Michael Woodroofe;Mary Meyer
估计单峰密度及其模式的问题引起了文献的广泛兴趣，从Barlow [1]，Prakasa Rao，[7]，Robertson，[8]，[8]和Wegman [13]，[14，[14）开始。并继续通过[9]，[2]，[3]，[5]和[4]，他们可以咨询以获取最大可能性估计量的进一步参考。该地区一些不一致在模式下，内核估计可以避免不一致，但必须在这里选择带宽。对数是对f的凹入，其对数的凹陷是岩石的意义[10] Polya频率函数提供了自己的模式的估计。模拟在第4节中报告了霍拉斯·拉克汉姆研究生院和国家科学基金会†National Science Foundation AMS AMS 2000主题支持分类：初级60K35，60K35;
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