  We extend balloon and sample-smoothing estimators, two types of
variable-bandwidth kernel density estimators, by a shift parameter and derive
their asymptotic properties. Our approach facilitates the unified study of a
wide range of density estimators which are subsumed under these two general
classes of kernel density estimators. We demonstrate our method by deriving the
asymptotic bias, variance, and mean (integrated) squared error of density
estimators with gamma, log-normal, Birnbaum-Saunders, inverse Gaussian and
reciprocal inverse Gaussian kernels. We propose two new density estimators for
positive random variables that yield properly-normalised density estimates.
Plugin expressions for bandwidth estimation are provided to facilitate easy
exploratory data analysis.
