Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval

We consider kernel-type methods for estimation of a density on [0, 1] which eschew explicit boundary correction. Our starting point is the successful implementation of beta kernel density estimators of Chen (1999). We propose and investigate two alternatives. For the first, we reverse the roles of estimation point x and datapoint Xi in each summand of the estimator. For the second, we provide kernels that are symmetric in x and X; these kernels are conditional densities of bivariate copulas. We develop asymptotic theory for the new estimators and compare them with Chen’s in a substantial simulation study. We also develop automatic bandwidth selection in the form of ‘rule-of-thumb’ bandwidths for all three estimators. We find that our second proposal, that based on ‘copula kernels’, seems particularly competitive with the beta kernel method of Chen in integrated squared error performance terms. Advantages include its greater range of possible values at 0 and 1, the fact that it is a bona fide density, and that the individual kernels and resulting estimator are comprehensible in terms of a direct single picture (as is ordinary kernel density estimation on the line).