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On distinguishability of two nonparametric sets of hypothesis

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  • Ermakov, M. S.

Abstract

Let we observe a signal S(t), t[set membership, variant](0,1) in Gaussian white noise [var epsilon] dw(t). The problem is to test a hypothesis S[set membership, variant][Theta]1[subset of]L2(0,1) versus alternatives S[set membership, variant][Theta]2[subset of]L2(0,1). The sets [Theta]1,[Theta]2 are closed and bounded. We show that there exists a statistical procedure allowing to make a true solution S[set membership, variant][Theta]1 or S[set membership, variant][Theta]2 with probability tending to one as [var epsilon]-->0 (i.e. to distinguish two nonparametric sets [Theta]1 and [Theta]2) iff there exists a finite-dimensional subspace H[subset of]L2(0,1) such that the projections [Theta]1 and [Theta]2 on H have no common points. A similar result is also obtained for the problems of testing hypotheses about density.

Suggested Citation

  • Ermakov, M. S., 2000. "On distinguishability of two nonparametric sets of hypothesis," Statistics & Probability Letters, Elsevier, vol. 48(3), pages 275-282, July.
    • Handle: RePEc:eee:stapro:v:48:y:2000:i:3:p:275-282
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