Abstract
We discuss a new weighted likelihood method for robust parametric estimation. The method is motivated by the need for generating a simple estimation strategy which provides a robust solution that is simultaneously fully efficient when the model is correctly specified. This is achieved by appropriately weighting the score function at each observation in the maximum likelihood score equation. The weight function determines the compatibility of each observation with the model in relation to the remaining observations and applies a downweighting only if it is necessary, rather than automatically downweighting a proportion of the observations all the time. This allows the estimators to retain full asymptotic efficiency at the model. We establish all the theoretical properties of the proposed estimators and substantiate the theory developed through simulation and real data examples. Our approach provides an alternative to the weighted likelihood method of Markatou et al. (J Stat Plan Inference 57(2):215–232, 1997; J Am Stat Assoc 93(442):740–750, 1998).
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References
Agostinelli C (2002a) Robust model selection in regression via weighted likelihood methodology. Stat Probab Lett 56(3):289–300
Agostinelli C (2002b) Robust stepwise regression. J Appl Stat 29(6):825–840
Agostinelli C (2007) Robust estimation for circular data. Comput Stat Data Anal 51(12):5867–5875
Agostinelli C, Greco L (2013) A weighted strategy to handle likelihood uncertainty in Bayesian inference. Comput Stat 28(1):319–339
Agostinelli C, Greco L (2018) Weighted likelihood estimation of multivariate location and scatter. Test 28(3):756–784. https://doi.org/10.1007/s11749-018-0596-0
Agostinelli C, Markatou M (1998) A one-step robust estimator for regression based on the weighted likelihood reweighting scheme. Stat Probab Lett 37(4):341–350
Agostinelli C, Markatou M (2001) Test of hypotheses based on the weighted likelihood methodology. Stat Sin 11:499–514
Basu A, Lindsay BG (1994) Minimum disparity estimation for continuous models: efficiency, distributions and robustness. Ann Inst Stat Math 46(4):683–705
Basu A, Harris IR, Hjort NL, Jones MC (1998) Robust and efficient estimation by minimising a density power divergence. Biometrika 85(3):549–559
Basu A, Shioya H, Park C (2011) Statistical inference: the minimum distance approach. Chapman & Hall/CRC, Boca Raton
Biswas A, Roy T, Majumder S, Basu A (2015) A new weighted likelihood approach. Stat 4(1):97–107
Field C, Smith B (1994) Robust estimation: a weighted maximum likelihood approach. Int Stat Rev 62(3):405–424
Gervini D, Yohai VJ (2002) A class of robust and fully efficient regression estimators. Ann Stat 30(2):583–616
Green PJ (1984) Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives. J R Stat Soc Ser B (Methodol) 46(2):149–192
Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35(1):73–101
Léger C, Romano JP (1990) Bootstrap choice of tuning parameters. Ann Inst Stat Math 42(4):709–735
Lehmann EL, Casella G (2006) Theory of point estimation. Springer, New York
Lenth RV, Green PJ (1987) Consistency of deviance-based \( M \)-estimators. J R Stat Soc Ser B Methodol 49(3):326–330
Lindsay BG (1994) Efficiency versus robustness: the case for minimum Hellinger distance and related methods. Ann Stat 22(2):1081–1114
Lubischew AA (1962) On the use of discriminant functions in taxonomy. Biometrics 18(4):455–477
Markatou M (2000a) A closer look at weighted likelihood in the context of mixtures. In: Charalambides CA, Koutras MV, Balakrishnan N (eds) Probability and statistical models with applications. CRC Press, Boca Raton, pp 447–467
Markatou M (2000b) Mixture models, robustness, and the weighted likelihood methodology. Biometrics 56(2):483–486
Markatou M, Basu A, Lindsay B (1997) Weighted likelihood estimating equations: the discrete case with applications to logistic regression. J Stat Plan Inference 57(2):215–232
Markatou M, Basu A, Lindsay B (1998) Weighted likelihood equations with bootstrap root search. J Am Stat Assoc 93(442):740–750
Montgomery DC, Peck EA, Vining GG (2012) Introduction to linear regression analysis, 5th edn. Wiley, New York
Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New York
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Staudte RG, Sheather SJ (1990) Robust estimation and testing. Wiley, New York
Stigler SM (1977) Do robust estimators work with real data? Ann Stat 5(6):1055–1098
Warwick J, Jones MC (2005) Choosing a robustness tuning parameter. J Stat Comput Simul 75(7):581–588
Woodruff RC, Mason JM, Valencia R, Zimmering S (1984) Chemical mutagenesis testing in Drosophila: I. Comparison of positive and negative control data for sex-linked recessive lethal mutations and reciprocal translocations in three laboratories. Environ Mutagen 6(2):189–202
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We gratefully acknowledge the comments received from the editor and the anonymous referees which greatly improved the presentation.
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Majumder, S., Biswas, A., Roy, T. et al. Statistical inference based on a new weighted likelihood approach. Metrika 84, 97–120 (2021). https://doi.org/10.1007/s00184-020-00778-y
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DOI: https://doi.org/10.1007/s00184-020-00778-y