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We extend balloon and sample-smoothing estimators, two types ofvariable-bandwidth kernel density estimators, by a shift parameter and derivetheir asymptotic properties. Our approach facilitates the unified study of awide range of density estimators which are subsumed under these two generalclasses of kernel density estimators. We demonstrate our method by deriving theasymptotic bias, variance, and mean (integrated) squared error of densityestimators with gamma, log-normal, Birnbaum-Saunders, inverse Gaussian andreciprocal inverse Gaussian kernels. We propose two new density estimators forpositive random variables that yield properly-normalised density estimates.Plugin expressions for bandwidth estimation are provided to facilitate easyexploratory data analysis.

Unified treatment of the asymptotics of asymmetric kernel density estimators