Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH: ∫π−π K{f(λ)} dλ=cagainstA: ∫π−π K{f(λ)} dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testTnbased on ∫π−π K{f(λ)} dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), 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(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.