Publication Date: October 2018
We derive bounds on the scope for a conﬁdence band to adapt to the unknown regularity of a nonparametric function that is observed with noise, such as a regression function or density, under the self-similarity condition proposed by Gine and Nickl (2010). We ﬁnd that adaptation can only be achieved up to a term that depends on the choice of the constant used to deﬁne self-similarity, and that this term becomes arbitrarily large for conservative choices of the self-similarity constant. We construct a conﬁdence band that achieves this bound, up to a constant term that does not depend on the self-similarity constant. Our results suggest that care must be taken in choosing and interpreting the constant that deﬁnes self-similarity, since the dependence of adaptive conﬁdence bands on this constant cannot be made to disappear asymptotically.
Keywords: Adaptation, Nonparametric inference, Self-similarity
JEL Classification Codes: C14
JEL Classification Codes: C14See CFDP Version(s): CFDP 2146R