Robust-efficient fitting of mixed linear models: Theory, simulations, actuarial extensions, and examples

In many areas of application mixed linear models serve as a popular tool for analyzing highly complex data sets. For inference about fixed effects and variance components, likelihood-based methods such as (restricted) maximum likelihood estimators, (RE)ML, are commonly pursued. However, it is well-known that these fully efficient estimators are extremely sensitive to small deviations from hypothesized normality of random components as well as to other violations of distributional assumptions. In this dissertation, we propose a new class of robust-efficient estimators for inference in mixed linear models. The new three-step estimation procedure provides truncated generalized least squares and variance components' estimators with hard-rejection weights adaptively computed from the data. Theoretical efficiency and robustness properties of the new estimators are established and then examined---via simulations---under a number of contaminating scenarios for small- and moderate-samples. Their trade-offs between efficiency and robustness are explored in comparison to well-established robust estimators, including robust (restricted) maximum likelihood, bounded influence estimators, Fellner's method, and constrained translated biweight S-estimators (CTBS). The detection of outlying data is discussed in detail. Widely studied real-data sets from chemistry and real estate serve to illustrate efficiency of detection rules and usefulness of new adaptively truncated likelihood (ATL) methods in practice. Further, we extend these procedures to some popular actuarial models. In particular, classical (regression) credibility models that can be embedded within the framework of mixed linear models are studied. In actuarial practice, it is well-known that standard and fully efficient estimators cannot be directly applied for skewed or long-tailed insurance data. Therefore, a second major objective of this dissertation is to develop robust and efficient methods for credibility when heavy-tailed claims are approximately log-location-scale distributed. To accomplish that, we first show how to express additive credibility models such as Buhlmann-Straub and Hachemeister as mixed linear models with symmetric or asymmetric errors. Then, we adjust adaptively truncated likelihood methods and compute highly robust credibility estimates for the ordinary but heavy-tailed claims part. Finally, we treat the identified excess claims separately and find robust-efficient credibility premiums. Monte Carlo simulations and case studies from property and casualty insurance and health care insurance are used to illustrate performance and functional capabilities of the new robust credibility estimators.