Summary
In the context of nonparametric density estimation, we consider the combination of automatic bandwidth selection rules with qualitative constraints on the search space of bandwidths derived from bounds on the number of modes. These constraints can be easily combined with an upper bound based on the concept of oversmoothing introduced by Terrell and Scott (1985).
Rather obviously, if a correct upper bound on the number of modes is known, our proposed approach helps to ensure an adequate representation of known qualitative features by the estimate. More surprisingly, even loose upper bounds on the number of modes are able to improve the MISE behavior of least squares cross-validation, by reducing the known tendency of under-smoothing of this bandwidth selector.
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Acknowledgement
The work has been supported by the Austrian Science Foundation FWF. We also wish to acknowledge a Research Infrastructure Grant from Murdoch University which helped in the initial support for collaboration leading to this article. The helpful comments of the referees are gratefully acknowledged.
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Futschik, A., Clarke, B.R. Improving bandwidth selection methods by adding qualitative constraints. CompStat 19, 445–453 (2004). https://doi.org/10.1007/BF03372106
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DOI: https://doi.org/10.1007/BF03372106