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Efficient and robust fitting of lognormal distributions. (English) Zbl 1084.62511

Summary: In parametric modeling of loss distributions in actuarial science, a versatile choice with intermediate tail weight is the lognormal distribution. Surprisingly, however, the fitting of this model using estimators which are at once efficient and robust has not been seriously addressed in the extensive literature. Consequently, for example, typical estimators of the lognormal mean and variance fail to be both efficient and robust. In particular, the highly efficient maximum likelihood estimators lack robustness. By robustness is meant limited sensitivity to outliers in the sample. For the two-parameter lognormal estimation problem, we consider equivalently the problem of efficient and robust joint estimation of the mean and variance of a normal model and introduce generalized median type estimators which are robust while also possessing very high efficiency compared to competitors already in the literature. These yield efficient and robust estimators of various parameters of interest in the lognormal model, and in this regard we provide detailed treatment of the lognormal mean. Extension of the approach to the much more complicated problem of estimation for the three-parameter lognormal model is also discussed.

MSC:

62F10 Point estimation
62F35 Robustness and adaptive procedures (parametric inference)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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