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Robust Estimation of the Tail Index of a Single Parameter Pareto Distribution from Grouped Data

Chudamani Poudyal

Abstract

Numerous robust estimators exist as alternatives to the maximum likelihood estimator (MLE) when a completely observed ground-up loss severity sample dataset is available. However, the options for robust alternatives to MLE become significantly limited when dealing with grouped loss severity data, with only a handful of methods like least squares, minimum Hellinger distance, and optimal bounded influence function available. This paper introduces a novel robust estimation technique, the Method of Truncated Moments (MTuM), specifically designed to estimate the tail index of a Pareto distribution from grouped data. Inferential justification of MTuM is established by employing the central limit theorem and validating them through a comprehensive simulation study.

Robust Estimation of the Tail Index of a Single Parameter Pareto Distribution from Grouped Data

Abstract

Numerous robust estimators exist as alternatives to the maximum likelihood estimator (MLE) when a completely observed ground-up loss severity sample dataset is available. However, the options for robust alternatives to MLE become significantly limited when dealing with grouped loss severity data, with only a handful of methods like least squares, minimum Hellinger distance, and optimal bounded influence function available. This paper introduces a novel robust estimation technique, the Method of Truncated Moments (MTuM), specifically designed to estimate the tail index of a Pareto distribution from grouped data. Inferential justification of MTuM is established by employing the central limit theorem and validating them through a comprehensive simulation study.
Paper Structure (5 sections, 4 theorems, 28 equations, 1 figure, 6 tables)

This paper contains 5 sections, 4 theorems, 28 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Pareto Grouped Data
  3. MTuM for Grouped Data
  4. Simulation Study
  5. Concluding Remarks

Key Result

Proposition 3.1

Suppose $1 \leq j\leq j' \leq m$. Then $\mathbb{C}ov \left( p_{j,n},1-p_{j',n} \right) = -\frac{p_{j}(1-p_{j'})}{n}$.

Figures (1)

  • Figure 3.1: Graphs of $g_{tT}$ for different values of $\theta$. Left panel represents the graph of $g_{tT}(\theta)$ for $\theta = 10$, $(t,T) = (2,12)$, and group boundaries vector $v_{1} = (0,5,10,15,20,25)$. Similarly, right panel represents the graph of $g_{tT}(\theta)$ for $\theta = 0.2$, $(t,T) = (.05,.45)$, and group boundaries vector $v_{2} = (0,.1,.2,.3,.4,.5)$.

Theorems & Definitions (7)

  • Proposition 3.1
  • proof
  • Corollary 3.1
  • Conjecture 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.1