### Abstract

Let {X_{t}} be a Gaussian ARMA process with spectral density f_{θ}(λ), where θ is an unknown parameter. The problem considered is that of testing a simple hypothesis H:θ = θ_{0} against the alternative A:θ ≠ θ_{0}. For this problem we propose a class of tests S, which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the χ^{2} type asymptotic expansion of the distribution of T ∈ S up to order n^{-1}, where n is the sample size. Also we derive the χ^{2} type asymptotic expansion of the distribution of T under the sequence of alternatives A_{n}: θ = θ_{0} + ε √n, ε{lunate} > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.

Original language | English |
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Pages (from-to) | 494-511 |

Number of pages | 18 |

Journal | Journal of Multivariate Analysis |

Volume | 27 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1988 Nov |

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### Keywords

- Gaussian ARMA process
- asymptotic expansion
- hypothesis testing
- maximum likelihood estimator
- power
- spectral density

### ASJC Scopus subject areas

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