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Robust Tail Index Estimation under Random Censoring via Minimum Density Power Divergence

Nour Elhouda Guesmia, Abdelhakim Necir, Djamel Meraghni

TL;DR

This work develops a robust tail-index estimator for Pareto-type tails under random right censoring by extending the minimum density power divergence (MDPD) framework to censored extreme value models. The proposed estimator $\hat{\gamma}_{1,\alpha}$ minimizes a Nelson–Aalen–based MDPD objective, achieving consistency under first-order regular variation with $p>1/2$ and asymptotic normality under a second-order framework with appropriate scaling. Through extensive simulations, the authors demonstrate strong robustness to contamination in the upper tail, with a tunable parameter $\alpha$ balancing efficiency and robustness, and they compare favorably against classical censored-tail estimators. Real-data applications to insurance losses and AIDS survival times illustrate practical benefits and limitations, notably under weak versus strong censoring, and motivate future extensions to handle extreme censoring regimes and covariates. Overall, the paper provides a principled, robust alternative for tail inference in censored data with clear theoretical guarantees and demonstrated empirical performance.

Abstract

We propose a robust estimator for the tail index of Pareto-type distributions under random right-censoring, based on the minimum density power divergence (MDPD) framework. To our knowledge, this is the first application of the MDPD approach to extreme value models with random censoring, opening a new direction for robust inference in this setting. Under mild regularity conditions, the estimator is shown to be consistent and asymptotically normal. Its performance in finite samples is extensively evaluated through simulation studies, demonstrating superior robustness and efficiency compared to existing methods. Contamination is introduced only before censoring to provide a meaningful assessment of robustness, while contamination after censoring is shown to yield distorted or unrealistic results. The practical relevance of the approach is illustrated using a real dataset on insurance claims, which features light censoring and fully observable extremes, and a dataset on AIDS survival times, which, despite stronger censoring (p<1/2), allows for illustrative comparisons and highlights practical limitations and challenges in more difficult scenarios.

Robust Tail Index Estimation under Random Censoring via Minimum Density Power Divergence

TL;DR

This work develops a robust tail-index estimator for Pareto-type tails under random right censoring by extending the minimum density power divergence (MDPD) framework to censored extreme value models. The proposed estimator minimizes a Nelson–Aalen–based MDPD objective, achieving consistency under first-order regular variation with and asymptotic normality under a second-order framework with appropriate scaling. Through extensive simulations, the authors demonstrate strong robustness to contamination in the upper tail, with a tunable parameter balancing efficiency and robustness, and they compare favorably against classical censored-tail estimators. Real-data applications to insurance losses and AIDS survival times illustrate practical benefits and limitations, notably under weak versus strong censoring, and motivate future extensions to handle extreme censoring regimes and covariates. Overall, the paper provides a principled, robust alternative for tail inference in censored data with clear theoretical guarantees and demonstrated empirical performance.

Abstract

We propose a robust estimator for the tail index of Pareto-type distributions under random right-censoring, based on the minimum density power divergence (MDPD) framework. To our knowledge, this is the first application of the MDPD approach to extreme value models with random censoring, opening a new direction for robust inference in this setting. Under mild regularity conditions, the estimator is shown to be consistent and asymptotically normal. Its performance in finite samples is extensively evaluated through simulation studies, demonstrating superior robustness and efficiency compared to existing methods. Contamination is introduced only before censoring to provide a meaningful assessment of robustness, while contamination after censoring is shown to yield distorted or unrealistic results. The practical relevance of the approach is illustrated using a real dataset on insurance claims, which features light censoring and fully observable extremes, and a dataset on AIDS survival times, which, despite stronger censoring (p<1/2), allows for illustrative comparisons and highlights practical limitations and challenges in more difficult scenarios.

Paper Structure

This paper contains 34 sections, 7 theorems, 145 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Minimum density power divergence
  3. MDPD estimator of $\gamma_{1}$ for right-censored data
  4. Consistency and asymptotic normality
  5. Simulation Study
  6. Data-generating mechanism
  7. Contamination schemes
  8. Generation of pseudo-samples from the Kaplan--Meier estimator
  9. Simulation scenarios and settings
  10. Selection of the number of upper order statistics
  11. Performance evaluation
  12. Simulation Results
  13. Contamination introduced "before" the censoring mechanism
  14. Scenario S1 (Burr distribution)
  15. Scenario S2 (Fréchet distribution)
  16. ...and 19 more sections

Key Result

Theorem 3.1

(Consistency) Assume that the cdfs $F$ and $G$ satisfy the first-order regular variation condition $\left( RV\right)$ and that $p>1/2.$ Let $k=k_{n}$ be a sequence of integers such that Then, for any $\alpha>0,$ with probability tending to $1,$ there exists a solution $\widehat{\gamma}_{1,\alpha}$ to the estimating equation $\left( estim-equa\right)$ such that

Figures (16)

  • Figure 9.1: Bias (top panel) and MSE (bottom panel) of the MDPD tail index estimator $\widehat{\gamma}_{1,\alpha},$ together with $\widehat{\gamma}_{1}^{(EFG)}$ (long-dashed purple line) and $\widehat{\gamma}_{1}^{(W)}$ (two-dashed orange line) computed from $2000$ Monte Carlo samples of size $1000$ genrated from Scenario S1, with $\gamma_{1}=0.3,$$\gamma_{c}=0.6$ and $p=0.55.$ Contamination is introduced before the censoring mechanism, with $\epsilon=0$ (left panel), $\epsilon=0.15$ (middle panel), and $\epsilon=0.40$ (right panel).
  • Figure 9.2: Bias (top panel) and MSE (bottom panel) of the MDPD tail index estimator $\widehat{\gamma}_{1,\alpha},$ together with $\widehat{\gamma}_{1}^{(EFG)}$ (long-dashed purple line) and $\widehat{\gamma}_{1}^{(W)}$ (two-dashed orange line) computed from $2000$ Monte Carlo samples of size $1000$ genrated from Scenario S1, with $\gamma_{1}=0.3,$$\gamma_{c}=0.6$ and $p=0.7.$ Contamination is introduced before the censoring mechanism, with $\epsilon=0$ (left panel), $\epsilon=0.15$ (middle panel), and $\epsilon=0.40$ (right panel).
  • Figure 9.3: Bias (top panel) and MSE (bottom panel) of the MDPD tail index estimator $\widehat{\gamma}_{1,\alpha},$ together with $\widehat{\gamma}_{1}^{(EFG)}$ (long-dashed purple line) and $\widehat{\gamma}_{1}^{(W)}$ (two-dashed orange line) computed from $2000$ Monte Carlo samples of size $1000$ genrated from Scenario S1, with $\gamma_{1}=0.5,$$\gamma_{c}=0.8$ and $p=0.55.$ Contamination is introduced before the censoring mechanism, with $\epsilon=0$ (left panel), $\epsilon=0.15$ (middle panel), and $\epsilon=0.40$ (right panel).
  • Figure 9.4: Bias (top panel) and MSE (bottom panel) of the MDPD tail index estimator $\widehat{\gamma}_{1,\alpha},$$\widehat{\gamma}_{1}^{(EFG)}$ (longdashed purple line) and $\widehat{\gamma}_{1}^{(W)}$ (twodashed orange line) based on $2000$ samples of size $1000$ from scenario S1, for $\gamma_{1}=0.5,$$\gamma_{c}=0.8$ and $p=0.7,$ under contamination introduced before the censoring mechanism, with $\epsilon=0$ (left panel), $\epsilon=0.15$ (middle panel) and $\epsilon=0.40$ (right panel).
  • Figure 9.5: Bias (top panel) and MSE (bottom panel) of the MDPD tail index estimator $\widehat{\gamma}_{1,\alpha},$ together with $\widehat{\gamma}_{1}^{(EFG)}$ (long-dashed purple line) and $\widehat{\gamma}_{1}^{(W)}$ (two-dashed orange line) computed from $2000$ Monte Carlo samples of size $1000$ genrated from Scenario S2, with $\gamma_{1}=0.3,$$\gamma_{c}=0.6$ and $p=0.55.$ Contamination is introduced before the censoring mechanism, with $\epsilon=0$ (left panel), $\epsilon=0.15$ (middle panel), and $\epsilon=0.40$ (right panel).
  • ...and 11 more figures

Theorems & Definitions (12)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 8.1
  • proof
  • Lemma 8.2
  • proof
  • Lemma 8.3
  • proof
  • Proposition 8.1
  • proof
  • ...and 2 more