Nonparametric approach for non-gaussian vector stationary processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrix f(λ). In this paper we consider the testing problem H:∫^π_-π K{f(λ)} dλ = c against A: ∫^π_-π K{f(λ)} dλ ≠ c, where K{·} is an appropriate function and c is a given constant. For this problem we propose a test T_n based on ∫^π_-π K{f̂_n(λ)} dλ, where f̂_n(λ) is a nonparametric spectral estimator of f(λ), and we define an efficacy of T_n under a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectra f^Z₄ of z(t). If it does not depend on f^Z₄, we say that T_n is non-Gaussian robust. We will give sufficient conditions for T_n to 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 of f(λ). The essential point of our approach is that we do not assume the parametric form of f(λ). Also some numerical studies are given and they confirm the theoretical results.