Likelihood-based prediction in models of ordered data [thesis]

Grigoriy Volovskiy, Udo Kamps, Marco Burkschat, Haikady N. Nagaraja
The main focus of the dissertation is on point prediction in models of ordered data comprised by the concept of generalized order statistics introduced by Kamps (1995). In particular, we study and apply prediction methods arising from applying the maximum likelihood principle to the well-known predictive likelihood function as well as to the observed predictive likelihood function, which has attracted little research interest as a tool for prediction. Interestingly, the conditional density
more » ... ion, which models the data generation process given the unknown distributional parameters as well as the value of the unobserved random quantity, and which algebraically produces the observed predictive likelihood function, naturally appears in the discussion of prediction sufficiency. By establishing the existence and uniqueness of the maximum likelihood predictor (MLP) of a future generalized order statistic based on a multiply Type-II censored sample of generalized order statistics from exponential distributions, we unify and extended several known results. In view of the generality of the assumed censoring scheme, the problem of maximum likelihood prediction of generalized order statistics from exponential distributions can be considered completely solved. Moreover, using several new asymptotic results for central generalized order statistics, which are also derive in the thesis, and that are of interest in themselves, we establish several asymptotic properties of the MLP such as strong consistency, asymptotic normality and asymptotic efficiency. In addition, a comparison in terms of the mean squared error of the MLP with the best linear unbiased predictor (BLUP) based on a Type-II doubly censored sample is presented. Furthermore, it is established that the MLP, the BLUP and the best linear equivariant predictor based on a general multiply Type-II censored sample all three are asymptotically efficient. The applicability and usefulness of the observed predictive likelihood function is demonstrated in the context of pred [...]
doi:10.18154/rwth-2018-231538 fatcat:mjaxas7aqnaj7p5tydm3kwp6yq