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The Extended Bregman Divergence and Parametric Estimation in Continuous Models

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  • Sancharee Basak

    (St. Xavier’s College, Kolkata)

  • Ayanendranath Basu

    (Indian Statistical Institute)

Abstract

Under standard regularity conditions, the maximum likelihood estimator (MLE) is the most efficient estimator at the model. However, modern practice recognizes that it is rare for the hypothesized model to hold exactly, and small departures from it are never entirely unexpected. But classical estimators like the MLE are extremely sensitive to the presence of noise in the data. Within the class of robust estimators, which constitutes parametric inference techniques designed to overcome the problems due to model misspecification and noise, minimum distance estimators have become quite popular in recent times. In particular, density-based distances under the umbrella of the Bregman divergence have been demonstrated to have several advantages. Here we will consider an extension of the ordinary Bregman divergence, and investigate the scope of parametric estimation under continuous models using this extended divergence proposal. Many of our illustrations will be based on the GSB divergence, a particular member of the extended Bregman divergence, which appears to hold high promise within the robustness area. To establish the usefulness of the proposed minimum distance estimation procedure, we will provide detailed theoretical investigations followed by substantial numerical verifications.

Suggested Citation

  • Sancharee Basak & Ayanendranath Basu, 2024. "The Extended Bregman Divergence and Parametric Estimation in Continuous Models," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(2), pages 333-365, November.
    • Handle: RePEc:spr:sankhb:v:86:y:2024:i:2:d:10.1007_s13571-024-00333-z
    DOI: 10.1007/s13571-024-00333-z
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    References listed on IDEAS

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    1. Ayanendranath Basu & Bruce Lindsay, 1994. "Minimum disparity estimation for continuous models: Efficiency, distributions and robustness," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(4), pages 683-705, December.
    2. Abhik Ghosh & Ayanendranath Basu, 2017. "The minimum S-divergence estimator under continuous models: the Basu–Lindsay approach," Statistical Papers, Springer, vol. 58(2), pages 341-372, June.
    3. Taranga Mukherjee & Abhijit Mandal & Ayanendranath Basu, 2019. "The B-exponential divergence and its generalizations with applications to parametric estimation," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(2), pages 241-257, June.
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