Online Updated Learning for Extremiles via Parametric Quantile Estimation

Extremile, a recently introduced coherent risk measure, has demonstrated significant potential as a valuable tool in financial and actuarial applications. However, developing a real time estimator for Extremile in streaming data environments presents substantial computational challenges. These difficulties primarily stem from the inability to generate ordered samples in data streams characterized by single-pass processing and undefined terminal size. To overcome this limitation, we propose an efficient online updating algorithm for Extremile based on a parameterized quantile-based approach by using generalized lambda distribution. Theoretically, this methodology eliminates the need for repeated full dataset reprocessing while maintaining asymptotic property. Both simulation studies and real data analyses are conducted to evaluate the finite sample performance of the proposed methods.