On testing a hypothesis which is close to a simple hypothesis

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].