ABSTRACT
Crucial information about the structure of the underlying target density is not revealed by examining solely its values, and is only discerned via its derivatives. For example, the local minima/maxima are characterised as locations where the first derivative is identically zero and the Hessian matrix is positive/negative definite. So there is great interest in complementing the density estimators with the density derivative estimators. This chapter introduces estimators of the derivatives of the density function and their practical bandwidth selectors, focusing on the first and second derivatives. It sets up a mathematical framework for optimal bandwidth selection. The chapter presents automatic bandwidth selectors for density derivative estimation and summarizes their convergence rates. It focuses on obtaining explicit results for the case of a normal density, which are required to compute data-based selectors, and fills in the mathematical details of the considered density derivative estimators.
