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Description:
Let we observe a signal S(t), t ∈ (0, 1) in Gaussian white noise ∈ dw(t). The problem is to test a hypothesis S ∈ Θ1 ⊂ L2 (0, 1) versus alternatives S ∈ Θ2 ⊂ L2(0, 1). The sets Θ1, Θ2 are closed and bounded. We show that there exists a statistical procedure allowing to make a true solution S ∈ Θ1 or S ∈ Θ2 with probability tending to one as ∈ → 0 ( i.e. to distinguish two nonparametric sets Θ1 and Θ2) iff there exists a finite-dimensional subspace H ⊂ L2 (0, 1) such that the projections Θ1 and Θ2 on H have no common points. A similar result is also obtained for the problems of testing hypotheses about density.