Abstract
For several years there has been debate over the relative merits of loss and risk as measures of the performance of nonparametric density estimators. In the way that this debate has dealt with risk, it has largely ignored the fact that any practical bandwidth selection rule must produce a random bandwidth. Existing theory for risk of density estimators is almost invariably concerned with nonrandom bandwidths. In the present paper we examine two different definitions of risk, both of them appropriate to circumstances where the bandwidth is random. Arguments in favor of, and motivations for, each approach are presented, including formulation of appropriate decision-theoretic frameworks. It is shown that the two approaches can give diametrically opposite answers to the question of which of two competing bandwidths selection rules is superior. Technical results include some surprising Conclusions about the nonexistence of risks, and even of moments of some common data-driven bandwidths under the usual assumptions.
Original language | English (US) |
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Pages (from-to) | 133-147 |
Number of pages | 15 |
Journal | Journal of Nonparametric Statistics |
Volume | 4 |
Issue number | 2 |
DOIs | |
State | Published - Jan 1 1994 |
Bibliographical note
Funding Information:'University of Minnesota. 2Australian National University; CSZRO Division of Mathematics and Statistics. 3Universitt5 Catholique de Louvain; Limburgs University Center; University of North Carolina, Chapel Hill. Research partly supported by NFWO Belgium and by NSF Grant No. DMS-9203135
Keywords
- Bandwidth selection
- decision theory
- density estimation
- kernel
- loss
- risk
- smoothing parameter