相依数据下正半轴上密度导数的Gamma核估计
Gamma kernel estimation of the density derivative on the positive semi-axis by dependent data
摘要：
我们估计定义在$[0,\infty)$上的概率密度函数的导数。为此目的，我们选择了核估计类的非对称伽玛核函数。伽玛核的使用是富有成效的，因为它们是非负的，根据半轴上的位置改变它们的形状，并且对于广泛的一类密度具有良好的边界性质。我们找到了一个最佳的内核带宽的最小值的均方误差的依赖数据与强混合。该带宽不同于针对伽马核密度估计提出的带宽。为此，我们导出了密度导数的协方差，并推导出其上界。最后，将所得结果应用于一阶强混合自回归过程的情形。估计的准确性进行了检查的模拟研究。提供了基于独立和非独立数据的拟议估计数的比较。
英文摘要：
We estimate the derivative of a probability density function defined on $[0,\infty)$. For this purpose, we choose the class of kernel estimators with asymmetric gamma kernel functions. The use of gamma kernels is fruitful due to the fact that they are nonnegative, change their shape depending on the position on the semi-axis and possess good boundary properties for a wide class of densities. We find an optimal bandwidth of the kernel as a minimum of the mean integrated squared error by dependent data with strong mixing. This bandwidth differs from that proposed for the gamma kernel density estimation. To this end, we derive the covariance of derivatives of the density and deduce its upper bound. Finally, the obtained results are applied to the case of a first-order autoregressive process with strong mixing. The accuracy of the estimates is checked by a simulation study. The comparison of the proposed estimates based on independent and dependent data is provided.
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