### 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 |
---|---|

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

### Cite this

**Third-order asymptomic properties of a class of test statistics under a local alternative.** / Taniguchi, Masanobu.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Third-order asymptomic properties of a class of test statistics under a local alternative

AU - Taniguchi, Masanobu

PY - 1991

Y1 - 1991

N2 - Suppose that {Xi; i = 1, 2, ...,} is a sequence of p-dimensional random vectors forming a stochastic process. Let pn, θ(Xn), Xn ∈ Rnp, be the probability density function of Xn = (X1, ..., Xn) depending on θ ∈ Θ, where Θ is an open set of R1. We consider to test a simple hypothesis H : θ = θ0 against the alternative A : θ ≠ θ0. 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).

AB - Suppose that {Xi; i = 1, 2, ...,} is a sequence of p-dimensional random vectors forming a stochastic process. Let pn, θ(Xn), Xn ∈ Rnp, be the probability density function of Xn = (X1, ..., Xn) depending on θ ∈ Θ, where Θ is an open set of R1. We consider to test a simple hypothesis H : θ = θ0 against the alternative A : θ ≠ θ0. 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).

KW - asymptotic expansion

KW - Bartlett's adjustment

KW - Gaussian ARMA process

KW - higher-order asymptotics of tests

KW - local alternative

KW - nonlinear regression model

KW - third-order most powerful test

UR - http://www.scopus.com/inward/record.url?scp=0001638264&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001638264&partnerID=8YFLogxK

U2 - 10.1016/0047-259X(91)90081-C

DO - 10.1016/0047-259X(91)90081-C

M3 - Article

VL - 37

SP - 223

EP - 238

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -