Prediction of missing order statistics for generalized extreme value distribution

The prediction of missing order statistics for the Generalized Extreme Value distribution is investigated‎, ‎with a focus on the Fréchet‎, ‎Gumbel‎, ‎and Weibull subtypes governed by the shape parameter γ‎. ‎This paper first establishes a necessary and sufficient condition for the existence of conditional moments of order statistics based on the domain of γ‎‎. ‎We then derive predictors using three distinct methods‎: ‎the best unbiased predictor‎, ‎the conditional median predictor‎, ‎and the conditional average predictor‎. ‎A comprehensive simulation study reveals that the optimal method is distribution-dependent‎. ‎For the Gumbel distribution‎, ‎the best unbiased predictor provides superior accuracy and robustness‎. ‎For the Fréchet distribution‎, ‎the best unbiased predictor is unequivocally superior‎, ‎while the conditional average predictor and the conditional median predictor demonstrate significant bias and instability‎. ‎For the Weibull distribution‎, ‎the choice is critically sensitive to γ‎, ‎best unbiased predictor is preferred for milder shapes‎, ‎whereas the conditional average predictor is more robust for stronger shapes‎. ‎These findings provide a clear‎, ‎evidence-based framework for selecting the appropriate prediction method in extreme value analysis‎. ‎To demonstrate the performance of the proposed methods‎, ‎a numerical example utilizing a real-world data set is provided.

Keywords:

Extreme value distribution , Missing order statistics , Extreme value index , prediction