Adaptation in minimax nonparametric hypothesis testing for ellipsoids and Besov bodies
- Ingster, Yuri, I.
2010 Mathematics Subject Classification
- 62G10 62G20
- nonparametric hypotheses testing, minimax hypotheses testing, adaptive hypotheses testing, asymptotics of error probabilities
We observe an infinitely dimensional Gaussian random vector $x=xi+v$ where $xi$ is a sequence of standard Gaussian variables and $vin l_2$ is an unknown mean. Let $V_varepsilon(tau,rho_varepsilon)subset l_2$ be sets which correspond to $l_q$-ellipsoids %of the radiuses $R/varepsilon$ of power semi-axes $a_i=i^-sR/varepsilon$ with $l_p$-ellipsoid %of the radiuses $rho_varepsilon/varepsilon$ and of semi-axes $b_i=i^-rrho_varepsilon/varepsilon$ removed or to similar Besov bodies $B_q,t,s(R/varepsilon)$ with Besov bodies $B_p,h,r(rho_varepsilon/varepsilon)$ removed. Here $tau =(kappa,R)$ or $tau =(kappa,h,t,R), kappa=(p,q,r,s)$ are the parameters which define the sets $V_varepsilon$ for given radiuses $rho_varepsilonto 0$, $0
0, varepsilonto 0$ is asymptotical parameter. For the case $tau$ is known hypothesis testing problem $H_0: v=0$ versus alternatives $H_varepsilon,tau:vin V_varepsilon(tau, rho_varepsilon)$ have been considered by Ingster and Suslina  in minimax setting. It was shown that there is a partition of the set of $kappa$ on to regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of two main and some "boundary" types). Also there is essential dependence of the structure of asymptotically minimax tests on the parameter $kappa$ for the case of Gaussian asymptotics . In this paper we consider alternative $H_varepsilon,Gamma:vin V_varepsilon(Gamma)$ for sets $V_varepsilon(Gamma)= bigcup_tauin GammaV_varepsilon(tau,rho_varepsilon(tau))$. This corresponds to adaptive setting: $tau$ is unknown, $tauin Gamma$ for a compact $Gamma=Ktimes Delta, Delta=[c, C]subset R_+^1, Ksubset Xi_G_1cup Xi_G_2$ where $ Xi_G_2$ and $ Xi_G_2$ are regions of main tapes of Gaussian asymptotics . First the problems of such types were studied by Spokoiny [16, 17]. For ellipsoidal case we study sharp asymptotics of minimax second kind errors $beta_varepsilon(alpha, Gamma)=beta(alpha, V_varepsilon(Gamma))$ and construct asymptotically minimax tests. % $psi_alpha,varepsilon,Gamma$. These asymptotics are analogous to degenerate type. For Besov bodies case we obtain exact rates and construct minimax consistent tests. Analogous exact rates are obtained in a signal detection problem for continuous variant of white Gaussian noise model: alternatives correspond to Besov or Sobolev balls with Sobolev or Besov balls removed. The study is based on results  and on an extension of methods of this paper for degenerate case.