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M-estimation in Multistage Sampling Procedures

Author

Listed:
  • Atul Mallik

    (Wells Fargo Securities)

  • Moulinath Banerjee

    (University of Michigan)

  • George Michailidis

    (University of Florida)

Abstract

Multi-stage (designed) procedures, obtained by splitting the sampling budget suitably across stages, and designing the sampling at a particular stage based on information about the parameter obtained from previous stages, are often advantageous from the perspective of precise inference. We develop a generic framework for M-estimation in a multistage setting and apply empirical process techniques to develop limit theorems that describe the large sample behavior of the resulting M-estimates. Applications to change-point estimation, inverse isotonic regression, classification, mode estimation and cusp estimation are provided: it is typically seen that the multistage procedure accentuates the efficiency of the M-estimates by accelerating the rate of convergence, relative to one-stage procedures. The step-by-step process induces dependence across stages and complicates the analysis in such problems, which we address through careful conditioning arguments.

Suggested Citation

  • Atul Mallik & Moulinath Banerjee & George Michailidis, 2020. "M-estimation in Multistage Sampling Procedures," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 261-309, August.
    • Handle: RePEc:spr:sankha:v:82:y:2020:i:2:d:10.1007_s13171-019-00194-z
    DOI: 10.1007/s13171-019-00194-z
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    References listed on IDEAS

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    1. Müller, Hans-Georg & Song, Kai-Sheng, 1997. "Two-stage change-point estimators in smooth regression models," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 323-335, June.
    2. Susan Wei & Michael R. Kosorok, 2013. "Latent Supervised Learning," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(503), pages 957-970, September.
    3. Bhattacharya, P.K., 1987. "Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 183-208, December.
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