Minimum Density Power Divergence Estimation for the Gamma Distribution with Applications to Robust Rainfall Modeling

2026-07-05

AI Summary

This study addresses the instability of traditional maximum likelihood estimation (MLE) in modeling rainfall data due to its sensitivity to outliers. Focusing on the gamma distribution, the authors develop a robust parameter estimation framework based on minimum density power divergence estimation (MDPDE). They derive explicit estimating equations and closed-form expressions for the asymptotic covariance matrix, and rigorously establish the robustness of the proposed estimator through influence function analysis. Comprehensive evaluation via asymptotic relative efficiency and Monte Carlo simulations demonstrates that MDPDE significantly outperforms MLE under contaminated data while retaining high efficiency in clean settings. Empirical application to Indian monsoon rainfall data containing outliers confirms that MDPDE achieves an effective balance between robustness and estimation efficiency.

Abstract

Statistical modeling of rainfall amounts is of considerable importance in meteorology, hydrology, and agriculture. The gamma distribution remains one of the most popular choices for modeling rainfall data due to its flexibility and ability to capture the skewness of rainfall observations. Rainfall datasets often contain atypical observations due to measurement errors and extreme weather events, making maximum likelihood estimation (MLE) highly sensitive to contamination. In this paper, we develop a robust estimation framework for the two-parameter gamma distribution based on the minimum density power divergence estimator (MDPDE). Explicit estimating equations are derived, and several theoretical properties of the proposed estimators are established. In particular, closed-form expressions for the asymptotic covariance matrix are obtained, and robustness is investigated through influence function analysis and asymptotic relative efficiency. The finite-sample performance of the estimators is examined through simulation studies under both pure and contaminated gamma models. The proposed methodology is further implemented to analyze detrended areally weighted monsoon rainfall data from the 36 meteorological subdivisions of India for the period 1951--2014. The results demonstrate that the MDPDE provides a useful compromise between robustness and efficiency, yielding more stable inference than MLE in the presence of outliers while maintaining high efficiency for uncontaminated data.

Problem

• robust estimation
• gamma distribution
• rainfall modeling
• outliers
• contaminated data

Innovation

• minimum density power divergence
• gamma distribution
• robust estimation
• influence function
• asymptotic covariance