Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

¹ St. Petersburg Transport University, Department of Applied Mathematics, Moskowskii Av. 9, 190031 St. Petersburg, Russia; ingster@pdmi.ras.ru.
² St. Petersburg Institute of Exact Mechanics and Optics, Technical University, Sablinskaya Str. 14, 197101 St. Petersburg, Russia; SuslinaI@mkk.ifmo.ru.

Received: 3 August 1998
Revised: 23 June 2000

Abstract

We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l₂ is an unknown mean. We consider the hypothesis testing problem H₀ : v = 0 versus alternatives for the sets . The sets V_ε are l_q-ellipsoids of semi-axes a_i = i^-s R/ε with l_p-ellipsoid of semi-axes b_i = i^-r p^ε/ε removed or similar Besov bodies B^q,t;s (R/ε) with Besov bodies B^p,h;r (p^ε/ε) removed. Here or are the parameters which define the sets V_ε for given radii p_ε → 0, 0 < p,q,h,t ≤ ∞, -∞ ≤ r,s ≤ ∞, R > 0; ε → 0 is the asymptotical parameter. We study the asymptotics of minimax second kind errors and construct asymptotically minimax or minimax consistent families of tests , if it is possible. We describe the partition of the set of parameters κ into regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of various types). Analogous rates have been obtained in a signal detection problem for continuous variant of white noise model: alternatives correspond to Besov or Sobolev balls with Besov or Sobolev balls removed. The study is based on an extension of methods of constructions of asymptotically least favorable priors. These methods are applicable to wide class of “convex separable symmetrical" infinite-dimensional hypothesis testing problems in white Gaussian noise model. Under some assumptions these methods are based on the reduction of hypothesis testing problem to convex extreme problem: to minimize specially defined Hilbert norm over convex sets of sequences of measures π_i on the real line. The study of this extreme problem allows to obtain different types of Gaussian asymptotics. If necessary assumptions do not hold, then we obtain other types of asymptotics.