### Abstract

Suppose that {X_{i}; i = 1, 2, ...,} is a sequence of p-dimensional random vectors forming a stochastic process. Let p_{n}, θ(X_{n}), X_{n} ∈ R^{np}, be the probability density function of X_{n} = (X_{1}, ..., X_{n}) depending on θ ∈ Θ, where Θ is an open set of R^{1}. We consider to test a simple hypothesis H : θ = θ_{0} against the alternative A : θ ≠ θ_{0}. For this testing problem we introduce a class of tests S, which contains the likelihood ratio, Wald, modified Wald, and Rao tests as special cases. Then we derive the third-order asymptotic expansion of the distribution of T ∈ S under a sequence of local alternatives. Using this result we elucidate various third-order asymptotic properties of T ∈ S (e.g., Bartlett's adjustments, third-order asymptotically most powerful properties). Our results are very general, and can be applied to the i.i.d. case, multivariate analysis, and time series analysis. Two concrete examples will be given. One is a Gaussian ARMA process (dependent case), and the other is a nonlinear regression model (non-identically distributed case).

Original language | English |
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Pages (from-to) | 223-238 |

Number of pages | 16 |

Journal | Journal of Multivariate Analysis |

Volume | 37 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 |

Externally published | Yes |

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### Keywords

- asymptotic expansion
- Bartlett's adjustment
- Gaussian ARMA process
- higher-order asymptotics of tests
- local alternative
- nonlinear regression model
- third-order most powerful test

### ASJC Scopus subject areas

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