Adaptive variable location kernel density estimators with good performance at boundaries

This paper introduces new adaptive versions of the variable location density estimator which, for the first time, achieve bias improvement by an order of magnitude at the boundaries, as well as affording the usual higher order bias in the interior of the density support. We develop a general theoretical framework into which both these and earlier versions of adaptive variable location density estimators fit. This enables us to provide a single formula for the higher order biases and variances of these estimators. Numerical work for the comparison of these estimators with each other and with the conventional kernel density estimator reveals good properties of the proposed estimators.