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Robust Estimation of Loss Models for Truncated and Censored Severity Data

Chudamani Poudyal, Vytaras Brazauskas

Abstract

In this paper, we consider robust estimation of claim severity models in insurance, when data are affected by truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance). In particular, robust estimators based on the methods of trimmed moments (T-estimators) and winsorized moments (W-estimators) are pursued and fully developed. The general definitions of such estimators are formulated and their asymptotic properties are investigated. For illustrative purposes, specific formulas for T- and W-estimators of the tail parameter of a single-parameter Pareto distribution are derived. The practical performance of these estimators is then explored using the well-known Norwegian fire claims data. Our results demonstrate that T- and W-estimators offer a robust and computationally efficient alternative to the likelihood-based inference for models that are affected by deductibles, policy limits, and coinsurance.

Robust Estimation of Loss Models for Truncated and Censored Severity Data

Abstract

In this paper, we consider robust estimation of claim severity models in insurance, when data are affected by truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance). In particular, robust estimators based on the methods of trimmed moments (T-estimators) and winsorized moments (W-estimators) are pursued and fully developed. The general definitions of such estimators are formulated and their asymptotic properties are investigated. For illustrative purposes, specific formulas for T- and W-estimators of the tail parameter of a single-parameter Pareto distribution are derived. The practical performance of these estimators is then explored using the well-known Norwegian fire claims data. Our results demonstrate that T- and W-estimators offer a robust and computationally efficient alternative to the likelihood-based inference for models that are affected by deductibles, policy limits, and coinsurance.
Paper Structure (28 sections, 76 equations, 2 figures)

This paper contains 28 sections, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Data and Models
  3. Complete Data
  4. Truncated Data
  5. Censored Data
  6. Insurance Payments
  7. T- and W-Estimation
  8. T-Estimators
  9. W-Estimators
  10. Asymptotic Properties
  11. T-Estimators
  12. W-Estimators
  13. Analytic Examples: Pareto I
  14. MLE
  15. Payments Y
  16. ...and 13 more sections

Figures (2)

  • Figure 1.1: Quantile functions of complete data and its trimmed and winsorized versions. Sample size: $n=50$. Trimming/winsorizing proportions: 10% (lower) and 20% (upper). Complete data marked by '$\circ$' and trimmed/winsorized by '$*$'.
  • Figure 5.1: Pareto quantile-quantile plots for the original and modified data sets. The dashed line represents the "best" fit line (in both cases): $y = 13.1 + 0.85 \, x$.