A Wavelet Method for Confidence Intervals in the Density Model

We present a wavelet procedure for defining confidence intervals for "f"("x"₀), where "x"₀ is a given point and "f" is an unknown density from which there are independent observations. We use an undersmoothing method which is shown to be near optimal (up to a logarithmic term) in a first order sense. We propose a second order correction using the Edgeworth expansion. The adaptation with respect to the unknown regularity of "f" is given via a Lepskii type algorithm and has the advantage to be well located. The theoretical results are proved under weak assumptions and concern very irregular or oscillating functions. An empirical study gives some hints for choosing the constant of the threshold level. The results are very encouraging for the length of the intervals as well as for the coverage accuracy. Copyright 2003 Board of the Foundation of the Scandinavian Journal of Statistics..