Sharp minimax tests for large covariance matrices and adaptation

摘要

We consider the detection problem of correlations in a $p$-dimensionalGaussian vector, when we observe $n$ independent, identically distributedrandom vectors, for $n$ and $p$ large. We assume that the covariance matrixvaries in some ellipsoid with parameter $\alpha >1/2$ and total energy boundedby $L>0$. We propose a test procedure based on a U-statistic of order 2 whichis weighted in an optimal way. The weights are the solution of an optimizationproblem, they are constant on each diagonal and non-null only for the $T$ firstdiagonals, where $T=o(p)$. We show that this test statistic is asymptoticallyGaussian distributed under the null hypothesis and also under the alternativehypothesis for matrices close to the detection boundary. We prove upper boundsfor the total error probability of our test procedure, for $\alpha>1/2$ andunder the assumption $T=o(p)$ which implies that $n=o(p^{ 2 \alpha})$. Weillustrate via a numerical study the behavior of our test procedure. Moreover,we prove lower bounds for the maximal type II error and the total errorprobabilities. Thus we obtain the asymptotic and the sharp asymptoticallyminimax separation rate $\tilde{\varphi} = (C(\alpha, L) n^2 p )^{- \alpha/(4\alpha + 1)}$, for $\alpha>3/2$ and for $\alpha >1$ together with theadditional assumption $p= o(n^{4 \alpha -1})$, respectively. We deduce rateasymptotic minimax results for testing the inverse of the covariance matrix. Weconstruct an adaptive test procedure with respect to the parameter $\alpha$ andshow that it attains the rate $\tilde{\psi}= ( n^2 p / \ln\ln(n\displaystyle\sqrt{p}) )^{- \alpha/(4 \alpha + 1)}$.