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Research Report SRR00-004

Chi-square oracle inequalities

Iain M. Johnstone

Abstract: We study soft threshold estimates of the non-centrality parameter \xi of a non-central \chid2(\xi) distribution, of interest, for example, in estimation of the squared length of the mean of a Gaussian vector. Mean squared error and oracle bounds, both upper and lower, are derived for all degrees of freedom d. These bounds are remarkably similar to those in the limiting Gaussian shift case. In nonparametric estimation of \int f2, a dyadic block implementation of these ideas leads to an alternate proof of the optimal adaptivity result of Efromovich and Low.


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