The Discrepancy Principle in Density Estimation

Details The Discrepancy Principle in Density Estimation The Discrepancy Principle in Density Estimation

2011-04-12

arxiv.org/abs/1104.2190v3

Mathematics

Statistics Theory

24 pages, 3 figures

Scientific paper

We investigate the discrepancy principle, a simple method for choosing smoothing parameters for nonparametric density estimation. The main idea is to maximally smooth subject to a constraint on the distance between the data and the estimate. This technique is one of the most widely known methods for choosing a regularization parameter in (deterministic) inverse problems, but has only rarely been applied in statistics. The most important exceptions originate in Statistical Learning Theory and the so-called Data Approximation approach. For the problem of density estimation, the discrepancy principle is based on a bound on the distance between the empirical and estimated distribution functions. We unify and extend previous results on kernel density estimation with bandwidths chosen by the discrepancy principle and derive analogous results for regular histograms. Most versions so far proposed asymptotically suffer from undersmoothing, since they force the integrated density estimate to lie too close to the empirical distribution function. We also show that for certain densities with infinite peaks using the discrepancy principle even leads to inconsistent estimators. Furthermore, we compare the discrepancy principle to standard methods in a simulation study. Surprisingly, although there are strong theoretical reasons for not using the criterion, some of the proposals work reasonably well over a large set of different densities and sample sizes, and the behavior of the methods at least up to n=2500 can be quite different from their asymptotic behavior.

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