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A continuous mapping theorem for the argmax‐functional in the non‐unique case

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  • Dietmar Ferger

Abstract

The Argmax‐Continuous Mapping Theorem (Argmax‐CMT) of Kim and Pollard resp. van derVaart and Wellner has been proved to be a very useful tool in statistics for deriving distributional convergence of M‐estimators. However it only works as long as the limit process possesses an almost sure unique maximizing point. In this article we prove an extension of the Argmax‐CMT where almost sure uniqueness is no longer needed. Moreover our Argmax‐CMT is also valid in the function space D(ℝ) equipped with Lindvall’s version of the Skorokhod‐topology. As an example the result is applied to change‐point estimators.

Suggested Citation

  • Dietmar Ferger, 2004. "A continuous mapping theorem for the argmax‐functional in the non‐unique case," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 58(1), pages 83-96, February.
    • Handle: RePEc:bla:stanee:v:58:y:2004:i:1:p:83-96
    DOI: 10.1046/j.0039-0402.2003.00111.x
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    Cited by:

    1. Habibi Reza, 2011. "Exact Distribution of Argmax (Argmin)," Stochastics and Quality Control, De Gruyter, vol. 26(2), pages 155-162, January.
    2. Cho, Haeran & Kirch, Claudia, 2022. "Bootstrap confidence intervals for multiple change points based on moving sum procedures," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    3. Eloyan, Ani & Ghosh, Sujit K., 2013. "A semiparametric approach to source separation using independent component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 383-396.
    4. Ferger Dietmar & Klotsche Jens, 2009. "Estimation of split-points in binary regression," Statistics & Risk Modeling, De Gruyter, vol. 27(02), pages 93-128, December.
    5. Lahkar, Ratul & Mukherjee, Sayan & Roy, Souvik, 2022. "Generalized perturbed best response dynamics with a continuum of strategies," Journal of Economic Theory, Elsevier, vol. 200(C).
    6. RatulLahkar & Sayan Mukherjee & Souvik Roy, 2021. "Generalized Perturbed Best Response Dynamics with a Continuum of Strategies," Working Papers 51, Ashoka University, Department of Economics.

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