Locally parametric nonparametric density estimation
Locally parametric nonparametric density estimation
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DOI：
10.1214/aos/1032298288
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发表时间：
1996-08
影响因子：
4.5
通讯作者：
N. Hjort;M. C. Jones
中科院分区：
文献类型：
--
作者：
N. Hjort;M. C. Jones
提出了一种具有参数泛音的非参数密度估计。假设f（x, θ）是某种密度族，由参数向量θ表示。我们定义了一个局部核平滑似然函数，对于每个x，可以用来估计真实密度的最佳局部参数近似值。这导致了f（x, o(x)）形式的新密度估计，从而为x的每个新值插入最佳局部参数估计。当使用的带宽很大时，这相当于普通的全似然参数密度估计，而对于中等和较小的带宽，该方法本质上是非参数的，仅使用数据和模型的局部属性。还描述了比通过局部似然更一般的替代方法。这些方法可以看作是在参数类中对参数进行非参数平滑的方法。研究了这种新的半参数估计量的性质。我们的首选版本具有与普通核方法大致相同的方差，但潜在的偏差较小。新方法在参数模型的非参数范围内优于传统的核方法，同时在模型正确的情况下，与全似然方法相比，新方法的精度不会损失太多。该方法的其他版本近似等同于在半参数框架中使用特定的高阶核。我们开发的方法可以看作是与非参数回归中的局部似然和局部加权最小二乘理论并行的密度估计。
This paper develops a nonparametric density estimator with parametric overtones. Suppose f(x, θ) is some family of densities, indexed by a vector of parameters θ. We define a local kernel-smoothed likelihood function which, for each x, can be used to estimate the best local parametric approximant to the true density. This leads to a new density estimator of the form f(x, o(x)), thus inserting the best local parameter estimate for each new value of x. When the bandwidth used is large, this amounts to ordinary full likelihood parametric density estimation, while for moderate and small bandwidths the method is essentially nonparametric, using only local properties of data and the model. Alternative ways more general than via the local likelihood are also described. The methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class. Properties of this new semiparametric estimator are investigated. Our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias. The new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed, while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct. Other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework. The methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression.