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Robust and Computationally Efficient Trimmed L-Moments Estimation for Parametric Distributions

Chudamani Poudyal, Qian Zhao, Hari Sitaula

TL;DR

This work addresses robust parametric estimation under contamination and heavy tails by developing a general framework of trimmed L-moments (MTM) that allows independent trimming For each moment. It derives closed-form MTM estimators for location-scale and Fréchet models, together with their asymptotic distributions via $L$-statistics theory and explicit variance expressions. Sign-selection rules resolve scale and tail-index ambiguities, and the approach is evaluated through simulation and a hurricane-loss real-data case, demonstrating improved robustness with competitive efficiency. Overall, MTM offers fast, stable, and adaptable inference for a wide range of distributions in the presence of outliers and heavy tails.

Abstract

This paper proposes a robust and computationally efficient estimation framework for fitting parametric distributions based on trimmed L-moments. Trimmed L-moments extend classical L-moment theory by downweighting or excluding extreme order statistics, resulting in estimators that are less sensitive to outliers and heavy tails. We construct estimators for both location-scale and shape parameters using asymmetric trimming schemes tailored to different moments, and establish their asymptotic properties for inferential justification using the general structural theory of L-statistics, deriving simplified single-integration expressions to ensure numerical stability. State-of-the-art algorithms are developed to resolve the sign ambiguity in estimating the scale parameter for location-scale models and the tail index for the Frechet model. The proposed estimators offer improved efficiency over traditional robust alternatives for selected asymmetric trimming configurations, while retaining closed-form expressions for a wide range of common distributions, facilitating fast and stable computation. Simulation studies demonstrate strong finite-sample performance. An application to financial claim severity modeling highlights the practical relevance and flexibility of the approach.

Robust and Computationally Efficient Trimmed L-Moments Estimation for Parametric Distributions

TL;DR

This work addresses robust parametric estimation under contamination and heavy tails by developing a general framework of trimmed L-moments (MTM) that allows independent trimming For each moment. It derives closed-form MTM estimators for location-scale and Fréchet models, together with their asymptotic distributions via -statistics theory and explicit variance expressions. Sign-selection rules resolve scale and tail-index ambiguities, and the approach is evaluated through simulation and a hurricane-loss real-data case, demonstrating improved robustness with competitive efficiency. Overall, MTM offers fast, stable, and adaptable inference for a wide range of distributions in the presence of outliers and heavy tails.

Abstract

This paper proposes a robust and computationally efficient estimation framework for fitting parametric distributions based on trimmed L-moments. Trimmed L-moments extend classical L-moment theory by downweighting or excluding extreme order statistics, resulting in estimators that are less sensitive to outliers and heavy tails. We construct estimators for both location-scale and shape parameters using asymmetric trimming schemes tailored to different moments, and establish their asymptotic properties for inferential justification using the general structural theory of L-statistics, deriving simplified single-integration expressions to ensure numerical stability. State-of-the-art algorithms are developed to resolve the sign ambiguity in estimating the scale parameter for location-scale models and the tail index for the Frechet model. The proposed estimators offer improved efficiency over traditional robust alternatives for selected asymmetric trimming configurations, while retaining closed-form expressions for a wide range of common distributions, facilitating fast and stable computation. Simulation studies demonstrate strong finite-sample performance. An application to financial claim severity modeling highlights the practical relevance and flexibility of the approach.
Paper Structure (10 sections, 7 theorems, 73 equations, 4 figures, 5 tables, 2 algorithms)

This paper contains 10 sections, 7 theorems, 73 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. General Method of Trimmed L-Moments
  3. Parametric Examples
  4. Location Scale Model
  5. Fréchet Model
  6. Simulation Study
  7. Real Data Analysis
  8. Conclusion
  9. Proofs
  10. Asymptotic Covariance Matrix Entries

Key Result

Theorem 1

With the trimming proportions satisfying inequality eqn:abCondition1, it follows that where

Figures (4)

  • Figure 1: Normal ARE curves under trimming inequalities \ref{['eqn:abCondition2']} or \ref{['eqn:abCondition3']}.
  • Figure 2: Frechet Densities. A larger $\beta$ (i.e., a smaller shape parameter $\alpha$) implies a heavier tail. This is because $S(x; \sigma, \beta_{1}) \le S(x; \sigma, \beta_{2})$ for all $x \ge 0$ and $\beta_{1} \le \beta_{2}$, where $S$ denotes the survival function.
  • Figure 3: Fréchet ARE curves under trimming inequalities \ref{['eqn:abCondition2']} or \ref{['eqn:abCondition3']}.
  • Figure 4: Empirical cumulative distribution function (CDF) of the hurricane damage data overlaid with fitted CDFs from the lognormal and Fréchet models, illustrating the diagnostic fit performance of each model.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2: Delta method
  • Proposition 1
  • proof
  • Corollary 1
  • Lemma 1
  • Corollary 2
  • proof
  • Corollary 3