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个局部对数多项式密度估计器，并作为一个定义良好的模型来消除多项式直方图的某些区域可能出现的负性问题。我们基于局部矩方法的估计方法可以看作是将相应的局部模型与局部样本矩相匹配的方法。我们称这个密度函数为“最大熵密度”。MED是一个“真正的”密度函数，即处处是非负的，且积分为1。我们证明了MED可以达到与多项式直方图相似的高阶收敛率。讨论了对数多项式密度函数的局部矩法与局部似然法的密切关系和区别。我们的方法可以看作是从聚合数据中估计连续密度函数的一种方法，可以在经济学等领域提供广泛的应用，在这些领域中，通常第一个可用的数据是预先分类或舍入的，以便于数据分析。

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.