Robust estimation of Pareto-type tail index through an exponential regression model

Abstract

In this paper, we introduce a robust estimator of the tail index of a Pareto-type distribution. The estimator is obtained through the use of the minimum density power divergence with an exponential regression model for log-spacings of top order statistics. The proposed estimator is compared to existing minimum density power divergence estimators of the tail index based on fitting an extended Pareto distribution and exponential regression model on log-ratio of spacings of order statistics. We derive the influence function and gross error sensitivity of the proposed estimator of the tail index to study its robustness properties. In addition, a simulation study is conducted to assess the performance of the estimators under different contaminated samples from different distributions. The results show that our proposed estimator of the tail index has better mean square errors and is less sensitive to an increase in the number of top order statistics. In addition, the estimation of the exponential regression model yields estimates of second-order parameters that can be used for estimation of extreme events such as quantiles and exceedance probabilities. The proposed estimator is illustrated with a practical dataset on insurance claims.

Estimation of the tail index of Pareto-type distributions using regularisation

“…First, the SOA Group Medical Insurance dataset consists of over 170,000 claims recorded from 1991 to 1992. In this study, we consider the 1991 dataset, which comprised 75,789 claims and have been studied widely in the extreme value context (see e.g [18], [4]). Considering the large size of this dataset, we focus on the extreme tail of the data and hence consider the top 10% data points, (i. e, 0 < k ≤ 0.1n).…”

“…Such large observations are suspected outliers and may significantly influence the tail index estimates (see e.g [4]). The convex curvature of the exponential Q-Q plots and the near linearity of the Pareto Q-Q plots of the datasets indicate that the datasets suggest that they belong to the Pareto-type distributions.…”

“…Pareto-type distributions are often encountered in applications in the area of Finance [1,2,3], re-insurance [4,5,6], risk management [7,8,9] and telecommunication [10,11]. This distribution type has tail function,…”

Estimation of the Tail Index of Pareto‐Type Distributions Using Regularisation

“…First, the SOA Group Medical Insurance dataset which consists of over 170,000 claims recorded from 1991 to 1992. In this study, we consider the 1991 dataset, which comprised 75,789 claims and have been studied widely in the extreme value context (see, for example, [4,18]). Considering the large size of this dataset, we focus on the extreme tail of the data and hence consider the top 10% data points, (i.e, 0 < k ≤ 0.1n).…”

“…Pareto-type distributions are often encountered in applications in the area of nance [1][2][3], reinsurance [4][5][6], risk management [7][8][9], and telecommunication [10,11]. is distribution type has tail function…”