Superefficiency in nonparametric function estimation. (English) Zbl 0895.62043
Summary: Fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems (e.g., nonparametric density or regression estimation). In this context several forms of superefficiency can occur. In contrast to what can happen in regular parametric problems, here every parameter point (e.g., unknown density or regression function) can be a point of superefficiency. We begin with an example which shows how fixed parameter asymptotic statements have often appeared in the study of adaptive kernel estimators, and how superefficiency can occur in this context. We then carry out a more systematic study of such fixed parameter statements. It is shown in four general settings how the degree of superefficiency attainable depends on the structural assumptions in each case.
MSC:
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62B15 | Theory of statistical experiments |
| 62M05 | Markov processes: estimation; hidden Markov models |
Keywords:
linear estimation; Sobolev parameter space; curve estimation; adaptive kernel estimators; superefficiencyReferences:
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| [26] | PHILADELPHIA, PENNSy LVANIA 19104 E-MAIL: lbrown@compstat.wharton.upenn.edu lowm@compstat.wharton.upenn.edu lzhao@compstat.wharton.upenn.edu |
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