### Abstract

For a class of locally stationary processes introduced by Dahlhaus, this paper discusses the problem of testing composite hypotheses. First, for the Gaussian likelihood ratio test (GLR), Wald test (W) and Lagrange multiplier test (LM), we derive the limiting distribution under a composite hypothesis in parametric form. It is shown that the distribution of GLR, W and LM tends to χ^{2} distribution under the hypothesis. We also evaluate their local powers under a sequence of local alternatives, and discuss their asymptotic optimality. The results can be applied to testing for stationarity. Some examples are given. They illuminate the local power property via simulation. On the other hand, we provide a nonparametric LAN theorem. Based on this result, we obtain the limiting distribution of the GLR under both null and alternative hypotheses described in nonparametric form. Finally, the numerical studies are given.

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

Pages (from-to) | 483-504 |

Number of pages | 22 |

Journal | Journal of Time Series Analysis |

Volume | 24 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2003 Jul |

Externally published | Yes |

### Fingerprint

### Keywords

- Gaussian likelihood ratio test
- Lagrange multiplier test
- Local asymptotic normality
- Local power
- Locally asymptotically optimal test
- Locally stationary processes
- Tests for stationarity
- Time-varying spectral density
- Transfer function
- Wald test

### ASJC Scopus subject areas

- Applied Mathematics
- Statistics and Probability

### Cite this

*Journal of Time Series Analysis*,

*24*(4), 483-504. https://doi.org/10.1111/1467-9892.00317

**Testing composite hypotheses for locally stationary processes.** / Sakiyama, Kenji; Taniguchi, Masanobu.

Research output: Contribution to journal › Article

*Journal of Time Series Analysis*, vol. 24, no. 4, pp. 483-504. https://doi.org/10.1111/1467-9892.00317

}

TY - JOUR

T1 - Testing composite hypotheses for locally stationary processes

AU - Sakiyama, Kenji

AU - Taniguchi, Masanobu

PY - 2003/7

Y1 - 2003/7

N2 - For a class of locally stationary processes introduced by Dahlhaus, this paper discusses the problem of testing composite hypotheses. First, for the Gaussian likelihood ratio test (GLR), Wald test (W) and Lagrange multiplier test (LM), we derive the limiting distribution under a composite hypothesis in parametric form. It is shown that the distribution of GLR, W and LM tends to χ2 distribution under the hypothesis. We also evaluate their local powers under a sequence of local alternatives, and discuss their asymptotic optimality. The results can be applied to testing for stationarity. Some examples are given. They illuminate the local power property via simulation. On the other hand, we provide a nonparametric LAN theorem. Based on this result, we obtain the limiting distribution of the GLR under both null and alternative hypotheses described in nonparametric form. Finally, the numerical studies are given.

AB - For a class of locally stationary processes introduced by Dahlhaus, this paper discusses the problem of testing composite hypotheses. First, for the Gaussian likelihood ratio test (GLR), Wald test (W) and Lagrange multiplier test (LM), we derive the limiting distribution under a composite hypothesis in parametric form. It is shown that the distribution of GLR, W and LM tends to χ2 distribution under the hypothesis. We also evaluate their local powers under a sequence of local alternatives, and discuss their asymptotic optimality. The results can be applied to testing for stationarity. Some examples are given. They illuminate the local power property via simulation. On the other hand, we provide a nonparametric LAN theorem. Based on this result, we obtain the limiting distribution of the GLR under both null and alternative hypotheses described in nonparametric form. Finally, the numerical studies are given.

KW - Gaussian likelihood ratio test

KW - Lagrange multiplier test

KW - Local asymptotic normality

KW - Local power

KW - Locally asymptotically optimal test

KW - Locally stationary processes

KW - Tests for stationarity

KW - Time-varying spectral density

KW - Transfer function

KW - Wald test

UR - http://www.scopus.com/inward/record.url?scp=0141976688&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141976688&partnerID=8YFLogxK

U2 - 10.1111/1467-9892.00317

DO - 10.1111/1467-9892.00317

M3 - Article

AN - SCOPUS:0141976688

VL - 24

SP - 483

EP - 504

JO - Journal of Time Series Analysis

JF - Journal of Time Series Analysis

SN - 0143-9782

IS - 4

ER -