# Likelihood-based prediction in models of ordered data

Volovskiy, Grigoriy; Kamps, Udo (Thesis advisor); Burkschat, Marco (Thesis advisor); Nagaraja, Haikady N. (Thesis advisor)

*Aachen (2018, 2019)* [Dissertation / PhD Thesis]

*Page(s): 1 Online-Ressource (vi, 247 Seiten) : Illustrationen*

Abstract

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 function, 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 predicting future record values. We establish the general form of the maximum observed likelihood predictor (MOLP) for continuous underlying distribution functions and apply this result to derive the MOLP for various distribution families such as exponential, extreme-value, Pareto, Lomax, Weibull, power function and uniform. We show that, in many cases, the MOLP is easy to derive and exhibits superior performance compared with MLP in terms of several commonly used performance metrics such as bias, mean squared error and Pitman’s measure of closeness. Moreover, it is shown that the MOLP naturally leads to a novel prediction method, the so-called maximum product of spacings prediction procedure, which is extendable to all models of generalized order statistics in a straightforward way. Based on the observation of the relation between the maximum observed likelihood prediction and the maximum product of spacings prediction procedures for record values, we introduce a novel family of prediction methods for generalized order statistics, so-called maximum entropy and minimum divergence predictors. This new prediction technique parallels several existing information theory-based techniques in parametric inference. We derive the form of the predictors for various particular instances of the family of prediction procedures as a function of the associated estimators of the distributional parameters as well as establish a general result concerning the form of a particular class of predictors, so-called maximum phi-entropy, resp. phi-divergence predictors. Finally, we use the result to predict future generalized order statistics from exponential distributions. Remarkably, one of the predictors coincides with the best linear unbiased predictor. In addition to point prediction, we briefly discuss prediction regions for future generalized order statistics from exponential distributions. We derive minimum size equivariant prediction regions for future generalized order statistics in the location, scale and location-scale set-ups based on a general multiply Type-II censored sample. All derived prediction regions turn out to be prediction intervals.

### Identifier

- REPORT NUMBER: RWTH-2018-231538