Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

We observe an infinitely dimensional Gaussian random vector x=#xi#+#upsilon# where #xi# is a sequence of standard Gaussian variables and #upsilon# element of l_2 is an unknown mean. We consider the hypothesis testing problem H_0: #upsilon#=0 versus alternatives H _e_l_e_m_e_n_t _o_f _,_#tau#: #upsilon# element of V _e_l_e_m_e_n_t _o_f for the sets V _e_l_e_m_e_n_t _o_f =V _e_l_e_m_e_n_t _o_f (#tau#, #rho# _e_l_e_m_e_n_t _o_f) is contained in l_2 which correspond to l_q-ellipsoids of the radiuses R/ element of and of power semi-axes a_i=i"-"s with l_p-ellipsoid of the radiuses #rho# _e_l_e_m_e_n_t _o_f / element of and of semi-axes b_i=i"-"r removed or to similar Besov bodies B_q_,_t_; _s(R/ element of) with Besov bodies B_p_,_h_; _r(#rho# _e_l_e_m_e_n_t _o_f / element of) removed. Here #tau#=(#kappa#, R) or #tau#=(#kappa#, h, t, R); #kappa#=(p, q, r, s) are the parameters which define the sets V _e_l_e_m_e_n_t _o_f for given radiuses #rho# _e_l_e_m_e_n_t _o_f #->#0, 0<p, q, h, t#<=##infinity#, -#infinity#<r, s <#infinity#R>0; element of #->#0 is the asymptotical parameter. We investigate the asymptotics of the minimax second kind errors probabilities #beta# _e_l_e_m_e_n_t _o_f (#alpha#)=#beta#(#alpha#, V _e_l_e_m_e_n_t _o_f (#tau#, #rho# _e_l_e_m_e_n_t _o_f)) and construct the asymptotical minimax or minimax consistent families of tests #psi#_#alpha#_; _e_l_e_m_e_n_t _o_f _,_#tau#_,_#rho# _e_l_e_m_e_n_t _o_f, if it is possible. We show that there are the division of the set of #kappa# on the regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of different types). The analogous rate asymptotics have been obtained in a signal detection problem for continuous variant of the white noise model: alternatives correspond to Besov or Sobolev balls with Sobolev or Besov balls removed. The study is based on an extension of methods of finding asymptotically least favorable priors. These methods are applicable to wide class of 'convex separable symmetrical' infinite-dimensional hypothesis testing problems for white Gaussian noise model. Under some assumptions these methods are based on the reduction of hypothesis testing problem to the convex extreme problem: to minimize special Hilbert norm over the convex sets of sequences anti #pi# of measures #pi#_i on the real line. Investigation of this extreme problem allows to obtain different types of Gaussian asymptotics. If necessary assumptions do not hold, then we obtain other types of the asymptotics. (orig.)Available from TIB Hannover: RR 3285(12)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman