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Asymptotics for estimation and testing procedures under loss of identifiability

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  • Zhu, Hongtu
  • Zhang, Heping
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    Abstract

    Statistical analyses commonly make use of models that suffer from loss of identifiability. In this paper, we address important issues related to the parameter estimation and hypothesis testing in models with loss of identifiability. That is, there are multiple parameter points corresponding to the same true model. We refer the set of these parameter points to as the set of true parameter values. We consider the case where the set of true parameter values is allowed to be very large or even infinite, some parameter values may lie on the boundary of the parameter space, and the data are not necessarily independently and identically distributed. Our results are applicable to a large class of estimators and their related testing statistics derived from optimizing an objective function such as a likelihood. We examine three specific examples: (i) a finite mixture logistic regression model; (ii) stationary ARMA processes; (iii) general quadratic approximation using Hellinger distance. The applications to these examples demonstrate the applicability of our results in a broad range of difficult statistical problems.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 97 (2006)
    Issue (Month): 1 (January)
    Pages: 19-45

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    Handle: RePEc:eee:jmvana:v:97:y:2006:i:1:p:19-45

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    Related research

    Keywords: Asymptotic distribution Cone Convergence rate Finite mixture models Hellinger distance Normal distribution;

    References

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    1. Andrews, Donald W K, 2001. "Testing When a Parameter Is on the Boundary of the Maintained Hypothesis," Econometrica, Econometric Society, vol. 69(3), pages 683-734, May.
    2. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-56, July.
    3. Hanfeng Chen & Jiahua Chen & John D. Kalbfleisch, 2001. "A modified likelihood ratio test for homogeneity in finite mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(1), pages 19-29.
    4. Donald W. K. Andrews, 1999. "Estimation When a Parameter Is on a Boundary," Econometrica, Econometric Society, vol. 67(6), pages 1341-1384, November.
    5. Andrews, Donald W.K., 1992. "Generic Uniform Convergence," Econometric Theory, Cambridge University Press, vol. 8(02), pages 241-257, June.
    6. Hanfeng Chen & Jiahua Chen & John D. Kalbfleisch, 2004. "Testing for a finite mixture model with two components," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 95-115.
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    Cited by:
    1. Pleydell, David R.J. & Chr├ętien, St├ęphane, 2010. "Mixtures of GAMs for habitat suitability analysis with overdispersed presence/absence data," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1405-1418, May.

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