Nonparametric gamma kernel estimators of density derivatives on positive semi-axis

Abstract

We consider nonparametric estimation of the derivative of a probability density function with the bounded support on [0, ∞). Estimates are looked up in the class of estimates with asymmetric gamma kernel functions. The use of gamma kernels is due to the fact they are nonnegative, change their shape depending on the position on the semi-axis and possess other good properties. We found analytical expressions for bias, variance, mean integrated squared error (MISE) of the derivative estimate. An optimal bandwidth, the optimal MISE, and rate of mean square convergence of the estimates for density derivative have also been found.

Gamma kernel estimation of the density derivative on the positive semi-axis by dependent data

“…It is shown that in the case of dependent data, assuming strong mixing, we can estimate the derivative of the pdf using the same technique that has been applied for independent data in [11]. Lemma 2.1, Section 2.1 contains the upper bound of the covariance.…”

“…To our best knowledge, the gamma kernels have been applied to the density derivative estimation at first time in [11]. In this paper the derivative f ′ (x) was estimated under the assumption that {X 1 , X 2 , .…”

“…This corresponds to the optimal bandwidth of order n − 1 7 for symmetrical kernels. The estimation of the univariate density derivative using a gamma kernel estimator by independent data was proposed in [11], [12]. This allows us to achieve the optimal MISE of the same order n −4/7 with a bandwidth of order n − 2 7 .…”