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Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters by the maximum likelihood and weighted score methods

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  • Ogasawara, Haruhiko

Abstract

In this paper, inverse expansions of parameter estimators are given in terms of their true values, where the estimators are obtained by the maximum likelihood and weighted score methods with constraints placed on the parameters using Lagrange multipliers. The corresponding expansions for estimated Lagrange multipliers are also given. These expansions are derived before and after studentization. The results with studentization give one-sided confidence intervals for the parameters up to third-order accuracy. As an application of the weighted score method, a modified Jeffreys prior to remove the asymptotic biases of the Lagrange multipliers as well as the parameter estimators is obtained under canonical parametrization in the exponential family.

Suggested Citation

  • Ogasawara, Haruhiko, 2016. "Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters by the maximum likelihood and weighted score methods," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 20-37.
    • Handle: RePEc:eee:jmvana:v:147:y:2016:i:c:p:20-37
    DOI: 10.1016/j.jmva.2015.12.015
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    References listed on IDEAS

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    1. Haruhiko Ogasawara, 2015. "Bias Adjustment Minimizing the Asymptotic Mean Square Error," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(16), pages 3503-3522, August.
    2. Ogasawara, Haruhiko, 2012. "Cornish-Fisher expansions using sample cumulants and monotonic transformations," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 1-18, January.
    3. Ogasawara, Haruhiko, 2010. "Asymptotic expansions for the pivots using log-likelihood derivatives with an application in item response theory," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2149-2167, October.
    4. Taniguchi, M. & Watanabe, Y., 1994. "Statistical Analysis of Curved Probability Densities," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 228-248, February.
    5. Takesi Hayakawa, 1977. "The likelihood ratio criterion and the asymptotic expansion of its distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 29(1), pages 359-378, December.
    6. Ogasawara, Haruhiko, 2015. "Expository supplement I to the paper“ Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters by the maximum likelihood and weighted score methods”," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 66(2/3), pages 9-58.
    7. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
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