Nonparametric Approach for Non-Gaussian Vector Stationary Processes

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

[1]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[2]  E. J. Hannan,et al.  Multiple time series , 1970 .

[3]  P. Bloomfield.  An exponential model for the spectrum of a scalar time series , 1973 .

[4]  Chris Chatfield,et al.  Introduction to Statistical Time Series. , 1976 .

[5]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

[6]  Douglas A. Wolfe,et al.  Introduction to the Theory of Nonparametric Statistics. , 1980 .

[7]  Masanobu Taniguchi,et al.  A Central Limit Theorem of Stationary Processes and the Parameter Estimation of Linear Processes (時系列解析の推測 : 理論と応用) , 1981 .

[8]  David R. Brillinger,et al.  Time Series: Data Analysis and Theory. , 1982 .

[9]  J. Geweke,et al.  Measurement of Linear Dependence and Feedback between Multiple Time Series , 1982 .

[10]  Masanobu Taniguchi,et al.  On estimation of the integrals of the fourth order cumulant spectral density , 1982 .

[11]  Ian Stewart,et al.  Complex Analysis , 1983 .

[12]  Marc Hallin,et al.  Linear serial rank tests for randomness against ARMA alternatives , 1984 .

[13]  Daniel M. Keenan,et al.  Limiting Behavior of Functionals of Higher-Order Sample Cumulant Spectra , 1987 .

[14]  Masanobu Taniguchi,et al.  Minimum Contrast Estimation for Spectral Densities of Stationary Processes , 1987 .

[15]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics (Revised Edition) , 1999 .

[16]  Karl-Rudolf Koch,et al.  Parameter estimation and hypothesis testing in linear models , 1988 .

[17]  Optimal rank-based procedures for time series analysis: testing an ARMA model against other ARMA models , 1988 .

[18]  NON-PARAMETRIC APPROACH IN TIME SERIES ANALYSIS , 1993 .