We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

我们使用未知密度函数f的对数的局部多项式逼近来估计密度及其导数。估计器保证是非负的，并在内部实现与f的支持边界上相同的最佳收敛速度. 因此，该估计器非常适合需要非负密度估计的应用，例如半参数最大似然估计。此外，我们表明我们的估计器在渐近性能和计算简便性方面都优于其他基于内核的方法。仿真结果证实，当这些替代方法与最佳输入（即 Epanechnikov 内核和最佳选择的带宽序列）一起使用时，与这些替代方法相比，我们的方法在有限样本中的表现相似或更好。我们提供多种语言的代码。