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Asymptotic normality of test statistics under alternative hypotheses

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  • Shapiro, Alexander

Abstract

The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics under alternative hypotheses. Second order asymptotics, and their implications for bias corrections, are also discussed in a somewhat informal manner. As an example, asymptotics of test statistics in the analysis of covariance structures are discussed in detail.

Suggested Citation

  • Shapiro, Alexander, 2009. "Asymptotic normality of test statistics under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 936-945, May.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:936-945
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    References listed on IDEAS

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    1. Alexander Shapiro, 1982. "Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(2), pages 187-199, June.
    2. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    3. Yuan, Ke-Hai & Hayashi, Kentaro & Bentler, Peter M., 2007. "Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1262-1282, July.
    4. McManus, Douglas A., 1991. "Who Invented Local Power Analysis?," Econometric Theory, Cambridge University Press, vol. 7(2), pages 265-268, June.
    5. R. Golden, 2003. "Discrepancy Risk Model Selection Test theory for comparing possibly misspecified or nonnested models," Psychometrika, Springer;The Psychometric Society, vol. 68(2), pages 229-249, June.
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    Cited by:

    1. Hao Wu & Michael Browne, 2015. "Random Model Discrepancy: Interpretations and Technicalities (A Rejoinder)," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 619-624, September.
    2. Po-Hsien Huang, 2017. "Asymptotics of AIC, BIC, and RMSEA for Model Selection in Structural Equation Modeling," Psychometrika, Springer;The Psychometric Society, vol. 82(2), pages 407-426, June.
    3. Chun, So Yeon & Alexander, Shapiro, 2009. "Normal versus Noncentral Chi-square Asymptotics of Misspecified Models," MPRA Paper 17310, University Library of Munich, Germany.

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