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Register a new userFunctional estimation and hypothesis testing in nonparametric boundary models
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Added by
Martin Wahl
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $int Phi(g(x)),dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Holder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $Phi$ which are satisfied for $Phi(u)=|u|^p$, $pge 1$. As an application, we consider the problem of estimating the $L^p$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.
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We discuss a general approach to handling multiple hypotheses testing in the case when a particular hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact set associated with the hypothesis. With our approach, this problem reduces to testing the hypotheses pairwise. Our central result is a test for a pair of hypotheses of the outlined type which, under appropriate assumptions, is provably nearly optimal. The test is yielded by a solution to a convex programming problem, so that our construction admits computationally efficient implementation. We further demonstrate that our assumptions are satisfied in several important and interesting applications. Finally, we show how our approach can be applied to a rather general detection problem encompassing several classical statistical settings such as detection of abrupt signal changes, cusp detection and multi-sensor detection.
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