Truncated, Censored, and Actuarial Payment-type Moments for Robust Fitting of a Single-parameter Pareto Distribution

TL;DR

This paper introduces a robust estimation method called the method of truncated moments for actuarial data, especially suited for heavy-tailed, censored, and truncated distributions, improving over traditional MLE approaches.

Contribution

It develops a new estimation technique, the method of truncated moments, tailored for actuarial models with truncation, censoring, and heavy tails, with theoretical and simulation validation.

Findings

The method of truncated moments outperforms MLE in robustness.

Asymptotic properties of the estimators are established.

Simulation studies confirm improved estimation accuracy.

Abstract

With some regularity conditions maximum likelihood estimators (MLEs) always produce asymptotically optimal (in the sense of consistency, efficiency, sufficiency, and unbiasedness) estimators. But in general, the MLEs lead to non-robust statistical inference, for example, pricing models and risk measures. Actuarial claim severity is continuous, right-skewed, and frequently heavy-tailed. The data sets that such models are usually fitted to contain outliers that are difficult to identify and separate from genuine data. Moreover, due to commonly used actuarial "loss control strategies" in financial and insurance industries, the random variables we observe and wish to model are affected by truncation (due to deductibles), censoring (due to policy limits), scaling (due to coinsurance proportions) and other transformations. To alleviate the lack of robustness of MLE-based inference in risk modeling, here in this paper, we propose and develop a new method of estimation - method of truncated moments (MTuM) and generalize it for different scenarios of loss control mechanism. Various asymptotic properties of those estimates are established by using central limit theory. New connections between different estimators are found. A comparative study of newly-designed methods with the corresponding MLEs is performed. Detail investigation has been done for a single parameter Pareto loss model including a simulation study.