On the Exact Asymptotic Behavior of Estimators of a Density and its Derivatives

Abstract

For an integer $p \geq 0$, Singh has proposed a class of kernel estimators $\hat{f}^{(p)}$ of the $p$th order derivative $f^{(p)}$ of a density $f$. This paper examines the detailed asymptotic behavior of these estimators. In particular, asymptotically equivalent expressions for the bias $(E\hat{f}^{(p)} - f^{(p)})$, the mean squared error $E(\hat{f}^{(p)} - f^{(p)})^2$ and the error $(\hat{f}^{(p)} - f^{(p)})$ are obtained, which in turn give exact rates of convergence of these terms to zero.