TY - JOUR

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

AU - Taniguchi, Masanobu

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1991/5

Y1 - 1991/5

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 - Bartlett's adjustment

KW - Gaussian ARMA process

KW - asymptotic expansion

KW - higher-order asymptotics of tests

KW - local alternative

KW - nonlinear regression model

KW - third-order most powerful test

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

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

M3 - Article

AN - SCOPUS:0001638264

VL - 37

SP - 223

EP - 238

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -