The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data

Abstract

Modeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries and risk managers often use such key risk indicators to determine the degree to which their companies are subject to particular aspects of risk, which arise from changes in underlying variables such as prices of equity, interest rates, or exchange rates. The present study proposes a new heavy-tailed exponential distribution that accommodates bathtub, upside-down bathtub, decreasing, decreasing-constant, and increasing hazard rates. Actuarial measures including value at risk, tail value at risk, tail variance, and tail variance premium are derived. A computational study for these actuarial measures is conducted, proving that the proposed distribution has a heavier tail as compared with the alpha power exponential, exponentiated exponential, and exponential distributions. We adopt six estimation approaches for estimating its parameters, and assess the performance of these estimators via Monte Carlo simulations. Finally, an actuarial real data set is analyzed, proving that the proposed model can be used effectively to model insurance data as compared with fifteen competing distributions.

A Flexible Extension of Pareto Distribution: Properties and Applications

“…This section discusses the conventional techniques for estimating the suggested model parameters θ =( a , b , α , β , k ) ⊤ by eight different classical estimation methods. Many papers discussed these methods (for more information, see [ 17 – 21 ]). Determining the estimated parameters in explicit form is mathematically complicated, so these estimates will be obtained numerically by using Wolfram Mathematica software version 12.0.…”

The Inverse-Power Logistic-Exponential Distribution: Properties, Estimation Methods, and Application to Insurance Data

“…We compare the proposed IPLE distribution with some other competing distributions, including the beta exponential (BE) [31], transmuted generalized exponential (TGE) [4], odd inverse Pareto exponential (OIPE) [16], exponentiated exponential (EE) [1], alpha power exponentiated exponential (APEE) [19], generalized odd log-logistic exponential (GLLE) [13], logistic exponential (LE) [21], alpha power exponential (APE) [9], Marshall-Olkin exponential (MOE) [32], Weibull (W), transmuted exponential (TE) [33], and exponential (E) distributions.…”

“…Hence, many researcher have been interested in proposing modified forms of the exponential distribution to increase its flexibility. Some recent extensions of the exponential distribution include the exponentiated exponential [1], beta exponential [2], beta generalized exponential [3], transmuted generalized exponential [4], Harris extended exponential [5], Kumaraswamy transmuted exponential [6], Marshall-Olkin Nadarajah-Haghighi [7], modified exponential [8], alpha power exponential [9,10], odd exponentiated half-logistic exponential [11], Marshall-Olkin logistic exponential [12], generalized odd log-logistic exponential [13], Marshall-Olkin alpha power exponential [14], extended odd Weibull exponential [15], odd inverse Pareto exponential [16], modified Kies exponential [17], Topp-Leone moment exponential [18], heavy-tailed exponential [19], and odd log-logistic Lindley exponential distributions [20].…”

The Weibull Generalized Exponential Distribution with Censored Sample: Estimation and Application on Real Data

“…e authors in [7] addressed the estimation of WGED parameters based on generalized order statistics, and they derived the submodels of generalized order statistics such as order statistics and record values. e authors in [8] introduced a heavy tailed exponential distribution by using the alpha power method for generalized continuous distribution. Teamah et al [9] presented Fréchet-Weibull mixture exponential distribution along with a variety of statistical properties.…”