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Consistency of M-estimators for non-identically distributed data: the case of fixed-design distributional regression

Author

Listed:
  • Bücher, Axel
  • Segers, Johan

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Staud, Torben

Abstract

This paper explores strong and weak consistency of M-estimators for non-identically distributed data, extending prior work. Emphasis is given to scenarios where data is viewed as a triangular array, which encompasses distributional regression models with non-random covariates. Primitive conditions are established for specific applications, such as estimation based on minimizing empirical proper scoring rules or conditional maximum likelihood. A key motivation is addressing challenges in extreme value statistics, where parameter-dependent supports can cause criterion functions to attain the value −∞, hindering the application of existing theorems.

Suggested Citation

  • Bücher, Axel & Segers, Johan & Staud, Torben, 2025. "Consistency of M-estimators for non-identically distributed data: the case of fixed-design distributional regression," LIDAM Discussion Papers ISBA 2025021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2025021
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    References listed on IDEAS

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    1. A. Dawid, 2007. "The geometry of proper scoring rules," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(1), pages 77-93, March.
    2. John H. J. Einmahl & Laurens Haan & Chen Zhou, 2016. "Statistics of heteroscedastic extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 31-51, January.
    3. Liese, F. & Vajda, I., 1994. "Consistency of M-Estimates in General Regression Models," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 93-114, July.
    4. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
    5. A. Philip Dawid & Monica Musio & Laura Ventura, 2016. "Minimum Scoring Rule Inference," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 123-138, March.
    6. Padoan, S. A. & Ribatet, M. & Sisson, S. A., 2010. "Likelihood-Based Inference for Max-Stable Processes," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 263-277.
    7. Gut, Allan, 1992. "The weak law of large numbers for arrays," Statistics & Probability Letters, Elsevier, vol. 14(1), pages 49-52, May.
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