Gamma Kernel Estimation of the Density Derivative on the Positive Semi-Axis by Dependent Data

Abstract

We estimate the derivative of a probability density function defined on [0,∞). 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.