This article is cited in
2 papers
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. Received: 16.03.1998
DOI:
10.4213/tvp468
Fulltext:
PDF file (710 kB)