Adaptive detection of high-dimensional signal

Let n-dimensional Gaussian random vector x = ξ + v be observed where ξ is a standard n-dimensional Gaussian vector and v ∈ Rn is the unknown mean. In the papers [3,5] there were studied minimax hypothesis testing problems: to test null - hypothesis H0 : v = 0 against two types of alternatives H1 = H1(θn): v ∈ Vn(θn). The first one corresponds to multi-channels signal detection problem for given value b of a signal and number k of channels containing a signal, θn = (b,k). The second one corresponds to lnq-ball of radius R1,n with the lnp-ball of radius R2,n removed, θn = (R1,n, R2,n,p,q) ∈ R4+. It was shown in [3,5] that often there are essential dependences of the structure of asymptotically minimax tests and of the asymptotics of the minimax second kind errors on parameters θn. These imply the problem: to construct adaptive tests having good minimax property for large enough regions Θn of parameters θn. This problem is studied here. We describe the sets Θn such that adaptation is possible without loss of efficiency. For other sets we present wide enough class of asymptotically exact bounds of adaptive efficiency and construct asymptotically minimax test procedures