### Abstract

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^{Z}
_{4} of z(t). If it does not depend on f^{Z}
_{4}, 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.

Original language | English |
---|---|

Pages (from-to) | 259-283 |

Number of pages | 25 |

Journal | Journal of Multivariate Analysis |

Volume | 56 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1996 Feb |

Externally published | Yes |

### Fingerprint

### Keywords

- Analysis of time series
- Efficacy
- Fourth-order cumulant spectral density
- Measure of linear dependence
- Non-Gaussian robustness
- Non-Gaussian vector stationary process
- Nonparametric hypothesis testing
- Principal components
- Spectral density matrix

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Numerical Analysis
- Statistics and Probability

### Cite this

*Journal of Multivariate Analysis*,

*56*(2), 259-283. https://doi.org/10.1006/jmva.1996.0014

**Nonparametric approach for non-gaussian vector stationary processes.** / Taniguchi, Masanobu; Puri, Madan L.; Kondo, Masao.

Research output: Contribution to journal › Article

*Journal of Multivariate Analysis*, vol. 56, no. 2, pp. 259-283. https://doi.org/10.1006/jmva.1996.0014

}

TY - JOUR

T1 - Nonparametric approach for non-gaussian vector stationary processes

AU - Taniguchi, Masanobu

AU - Puri, Madan L.

AU - Kondo, Masao

PY - 1996/2

Y1 - 1996/2

N2 - 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 Tn based on ∫π -π K{f̂n(λ)} dλ, where f̂n(λ) is a nonparametric spectral estimator of f(λ), and we define an efficacy of Tn under a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectra fZ 4 of z(t). If it does not depend on fZ 4, we say that Tn is non-Gaussian robust. We will give sufficient conditions for Tn 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.

AB - 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 Tn based on ∫π -π K{f̂n(λ)} dλ, where f̂n(λ) is a nonparametric spectral estimator of f(λ), and we define an efficacy of Tn under a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectra fZ 4 of z(t). If it does not depend on fZ 4, we say that Tn is non-Gaussian robust. We will give sufficient conditions for Tn 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.

KW - Analysis of time series

KW - Efficacy

KW - Fourth-order cumulant spectral density

KW - Measure of linear dependence

KW - Non-Gaussian robustness

KW - Non-Gaussian vector stationary process

KW - Nonparametric hypothesis testing

KW - Principal components

KW - Spectral density matrix

UR - http://www.scopus.com/inward/record.url?scp=0030077092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030077092&partnerID=8YFLogxK

U2 - 10.1006/jmva.1996.0014

DO - 10.1006/jmva.1996.0014

M3 - Article

AN - SCOPUS:0030077092

VL - 56

SP - 259

EP - 283

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -