The main purpose of the present work is to investigate kernel-type estimate of a class of function derivatives including parameters such as the density, the conditional cumulative distribution function and the regression function. The uniform strong convergence rate is obtained for the proposed estimates and the central limit theorem is established under mild conditions. Moreover, we study the asymptotic mean integrated square error of kernel derivative estimator which plays a fundamental role in the characterization of the optimal bandwidth. The obtained results in this paper are established under a general setting of discrete time stationary and ergodic processes. A simulation study is performed to assess the performance of the estimate of the derivatives of the density function as well as the regression function under the framework of a discretized stochastic processes. An application to financial asset prices is also considered for illustration.

目前工作的主要目的是研究一类函数导数的核型估计，包括密度、条件累积分布函数和回归函数等参数。对于所提出的估计，获得了一致的强收敛率，并且在温和条件下建立了中心极限定理。此外，我们研究了核导数估计器的渐近平均积分平方误差，它在最佳带宽的表征中起着重要作用。本文得到的结果是在离散时间平稳和遍历过程的一般设置下建立的。进行模拟研究以评估在离散化随机过程的框架下对密度函数以及回归函数的导数的估计的性能。为了说明，还考虑了对金融资产价格的应用。