Characterizing saddle point approximations for tail probability estimation with applications to Teesta valley hydroelectric feasibility

Abstract

Between 2010 and 2022, the Teesta Valley in northeastern India recorded average annual discharge fluctuations exceeding 37.4%, making hydroelectric feasibility assessment dependent on reliable tail probability estimates. This paper develops a modified saddlepoint approximation framework for estimating tail probabilities of non-standard distributions encountered in hydrological flow data, and applies the method to evaluate hydroelectric generation potential across six monitoring stations in the Teesta basin. The proposed approach extends Lugannani-Rice saddlepoint formulas by embedding change-point detection within the cumulant generating function estimation, allowing the approximation to adapt when underlying flow distributions shift between monsoon and dry seasons. Partial likelihood methods were used to handle censored observations from periods when gauging equipment was submerged or damaged. An ANOVA-based module comparison tested whether saddlepoint corrections varied significantly across river segments. Continued fraction representations of the tail integral improved numerical stability for extreme quantiles beyond the 99th percentile. Results from six Teesta Valley stations showed that the modified saddlepoint method reduced mean absolute tail probability error from 0.0187 (normal approximation) to 0.0043 across all stations, with the largest gains at stations exhibiting bimodal flow distributions. The ANOVA confirmed that correction magnitudes differed across segments (F = 14.73, p<0.001). Change-point detection identified an average of 2.3 regime shifts per station over the twelve-year record, primarily associated with monsoon onset and withdrawal. These findings indicate that saddlepoint methods, when combined with structural break detection, provide markedly more accurate feasibility inputs than standard distributional assumptions for hydroelectric planning in monsoon-affected river systems.