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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 2, Pages 356–368 (Mi tvp468)

Short Communications

On testing a hypothesis which is close to a simple hypothesis

Yu. I. Ingster

Petersburg State Transport University

Abstract: Let an n-dimensional Gaussian vector $x=v+\xi$ be observed, where $v\in \mathbf{R}^n$ is an unknown vector of means and $\xi$ is a standard n-dimensional Gaussian vector. We consider, as $n\to\infty$, an asymptotically minimax problem of testing a hypothesis $H_{0}: \|v\|_p\le R_{n,0}$ against an alternative $H_{1}: \|v\|_p\ge R_{n,1}$. It is known [Yu. I. Ingster, Math. Methods Statist., 2 (1993), pp. 85–114; 171–189; 249–268] that if $H_0$ is simple (that is, if $R_{n,0}=0$), the conditions of minimax distinguishability or indistinguishability have the form $R_{n,1}/R^*_{n,1,p_{\vphantom{2}}}\to\infty$, $R_{n,1}/R^*_{n,1,p_{\vphantom{2_a}}}\to 0$, respectively, and are expressed in terms of the critical radii $R^*_{n,1,p}$. We are interested in the problem of how small $R_{n,0}$ can be to keep these conditions of distinguishability and indistinguishability.
The solution has the form $R_{n,0}=o(R^*_{n,0,p})$ and is expressed in terms of the critical radii $R^*_{n,0,p_{\vphantom{2_a}}}$, the form of which depends on the evenness of $p$. In particular, the exponent of the critical radii $R^*_{n,0,p_{\vphantom{2_a}}}$ has, as a function of $p$, a discontinuity to the left for even $p > 2$; in addition, $R^*_{n,0,p}\asymp R^*_{n,1,p}$ only if $p$ is even. These results are transferred to the model corresponding to observations over an unknown signal $f$ from a Sobolev or Besov class in a Gaussian white noise.
Recently, analogous phenomena in the problem of estimating the functional $\Phi(f)=\|f\|_p$ have been established in [O. V. Lepski, A. Nemirovski, and V. G. Spokoiny, Probab. Theory Related Fields, 113 (1999), pp. 221–253].

Keywords: minimax hypothesis testing, nonparametric hypotheses and alternatives, Sobolev and Besov classes.

DOI: https://doi.org/10.4213/tvp468

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English version:
Theory of Probability and its Applications, 2001, 45:2, 310–323

Bibliographic databases:

Citation: Yu. I. Ingster, “On testing a hypothesis which is close to a simple hypothesis”, Teor. Veroyatnost. i Primenen., 45:2 (2000), 356–368; Theory Probab. Appl., 45:2 (2001), 310–323

Citation in format AMSBIB
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• https://doi.org/10.4213/tvp468
• http://mi.mathnet.ru/eng/tvp/v45/i2/p356

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This publication is cited in the following articles:
1. Juditsky A., Nemirovski A., “On nonparametric tests of positivity/monotonicity/convexity”, Annals of Statistics, 30:2 (2002), 498–527
2. “To the memory of Yu. I. Ingster”, J. Math. Sci. (N. Y.), 204:1 (2015), 1–6
3. Blanchard G. Carpentier A. Gutzeit M., “Minimax Euclidean Separation Rates For Testing Convex Hypotheses in R-D”, Electron. J. Stat., 12:2 (2018), 3713–3735
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