A study on statistical theory of density estimation via the Bernstein polynomial and its application

在这项研究中，我们集中于通过Bernstein多项式逼近来估计函数的非参数方法，例如概率密度(独立且同分布的环境)或谱密度(平稳过程)。我们还致力于开发使用一些非对称核的方法。特别是，我们提出了(I)对基于Bernstein估计的加性或乘性(因此，非负)偏差修正，以及(Ii)一类修正的基于Bessel核的估计量。严格地推导了它们的渐近性质(偏差、方差、平均积分平方误差等)，包括渐近正态分布。此外，我们重新检查了(III)伽马核密度估计器，以获得更好的MISE性能。基于模拟数据集和真实数据集的几个数值研究证实了我们的研究结果。

In this study, we focus on the nonparametric methods via the Bernstein polynomial approximation to estimate a function such as the probability density (independent and identically distributed setting) or the spectral density (stationary process). We also aimed at developing methods using some asymmetric kernels. Especially, we have proposed (i) additive or multiplicative (hence, nonnegative) bias corrections to the Bernstein-based estimator and (ii) a class of modified Bessel kernel-based estimators. Their asymptotic properties (bias, variance, mean integrated squared error, and so on), including the asymptotic normality, were derived rigorously. Furthermore, we have re-examined (iii) the gamma kernel density estimator to get a better performance in terms of MISE. Several numerical studies based on both simulated and real data sets were presented to confirm our results in this study.