WIAS Preprint No. 419, (1998)

Adaptation in minimax nonparametric hypothesis testing for ellipsoids and Besov bodies



Authors

  • Ingster, Yuri, I.

2010 Mathematics Subject Classification

  • 62G10 62G20

Keywords

  • nonparametric hypotheses testing, minimax hypotheses testing, adaptive hypotheses testing, asymptotics of error probabilities

Abstract

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 $rho_varepsilonto 0$, $00, varepsilonto 0$ 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 [11] 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 [11] and on an extension of methods of this paper for degenerate case.

Download Documents