Method of Local Moments and Nonparametric Estimation of Maximum Entropy Density Function

Sagae和Scott（1997）基于最大熵原理提出了一种局部对数多项式密度估计器，并将其作为一个定义良好的模型，以消除多项式直方图某些区域可能出现的负值问题。我们的估计方法的基础上的局部矩的方法可以被看作是一种方法匹配相应的局部模型的局部样本矩。我们称这个密度函数为“最大熵密度（MED）”。MED是一个“真正的”密度函数，也就是说，在任何地方都是非负的，并且积分为1。证明了MED方法具有与多项式直方图相似的高阶收敛速度，并讨论了对数多项式密度函数的局部矩方法与局部似然方法之间的联系和区别。我们的方法可以被看作是一种从聚合数据中估计连续密度函数的方法，并且可以在诸如经济学等领域提供广泛的应用，在这些领域中，通常情况下，第一个可用的数据在数据分析之前被预先分箱或舍入。

A local log polynomial density estimator is proposed based on the maximum entropy principle and is also done as a well-defined model in order to remove the problem of negativity that can occur in certain region of the polynomial histogram by Sagae and Scott (1997). Our estimating approach based on the method of local moments might be viewed as a method matching the corresponding local model to the localized sample moments. We call this density function the "Maximum Entropy Density (MED)". The MED is a "bona fide" density function, that is, nonnegative everywhere with integral to one. We show that the MED can achive the higher order convergence rates parallel to the results for the polynomial histogram.We also discuss a close relation and the difference between the method of local moments and the local likelihood method for the log-polynomial density function. Our method can be seen as a way for estimating the continuous density function from the aggregated data and may provide a wide range of applications in fields such as economics where it is often the case that the first data available are pre-binned or rounded prior for the data analysis.