Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf([lambda]). In this paper we consider the testing problemH: [integral operator][pi]-[pi] K{f([lambda])} d[lambda]=cagainstA: [integral operator][pi]-[pi] K{f([lambda])} d[lambda][not equal to]c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testTnbased on [integral operator][pi]-[pi] K{f([lambda])} d[lambda]=c, wheref([lambda]) is a nonparametric spectral estimator off([lambda]), and we define an efficacy ofTnunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf4Zofz(t). If it does not depend onf4Z, we say thatTnis non-Gaussian robust. We will give sufficient conditions forTnto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off([lambda]). The essential point of our approach is that we do not assume the parametric form off([lambda]). Also some numerical studies are given and they confirm the theoretical results.