scholarly journals On Modeling the Earthquake Insurance Data via a New Member of the T-X Family

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Zubair Ahmad ◽  
Eisa Mahmoudi ◽  
Omid Kharazmi
Keyword(s):  
At Risk ◽  
Maximum Likelihood ◽  
Simulation Study ◽  
Value At Risk ◽  
Model Parameters ◽  
Earthquake Insurance ◽  
Monte Carlo Simulation Study ◽  
Heavy Tailed ◽  
Modeling Data ◽  
Insurance Data

Heavy-tailed distributions play an important role in modeling data in actuarial and financial sciences. In this article, a new method is suggested to define new distributions suitable for modeling data with a heavy right tail. The proposed method may be named as the Z-family of distributions. For illustrative purposes, a special submodel of the proposed family, called the Z-Weibull distribution, is considered in detail to model data with a heavy right tail. The method of maximum likelihood estimation is adopted to estimate the model parameters. A brief Monte Carlo simulation study for evaluating the maximum likelihood estimators is done. Furthermore, some actuarial measures such as value at risk and tail value at risk are calculated. A simulation study based on these actuarial measures is also done. An application of the Z-Weibull model to the earthquake insurance data is presented. Based on the analyses, we observed that the proposed distribution can be used quite effectively in modeling heavy-tailed data in insurance sciences and other related fields. Finally, Bayesian analysis and performance of Gibbs sampling for the earthquake data have also been carried out.

scholarly journals The Arcsine Exponentiated-X Family: Validation and Insurance Application

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
Wenjing He ◽  
Zubair Ahmad ◽  
Ahmed Z. Afify ◽  
Hafida Goual
Keyword(s):  
At Risk ◽  
Weibull Distribution ◽  
Value At Risk ◽  
Claims Data ◽  
Numerical Study ◽  
Model Parameters ◽  
Insurance Claims ◽  
Goodness Of Fit Test ◽  
Insurance Claims Data ◽  
Heavy Tailed

In this paper, we propose a family of heavy tailed distributions, by incorporating a trigonometric function called the arcsine exponentiated-X family of distributions. Based on the proposed approach, a three-parameter extension of the Weibull distribution called the arcsine exponentiated-Weibull (ASE-W) distribution is studied in detail. Maximum likelihood is used to estimate the model parameters, and its performance is evaluated by two simulation studies. Actuarial measures including Value at Risk and Tail Value at Risk are derived for the ASE-W distribution. Furthermore, a numerical study of these measures is conducted proving that the proposed ASE-W distribution has a heavier tail than the baseline Weibull distribution. These actuarial measures are also estimated from insurance claims real data for the ASE-W and other competing distributions. The usefulness and flexibility of the proposed model is proved by analyzing a real-life heavy tailed insurance claims data. We construct a modified chi-squared goodness-of-fit test based on the Nikulin–Rao–Robson statistic to verify the validity of the proposed ASE-W model. The modified test shows that the ASE-W model can be used as a good candidate for analyzing heavy tailed insurance claims data.

scholarly journals Heavy-Tailed Log-Logistic Distribution: Properties, Risk Measures and Applications

2021 ◽  
Vol 9 (4) ◽  
pp. 910-941
Author(s):  
Abd-Elmonem A. M. Teamah ◽  
Ahmed A. Elbanna ◽  
Ahmed M. Gemeay
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Real Data ◽  
Estimation Methods ◽  
Logistic Distribution ◽  
Model Parameters ◽  
Data Sets ◽  
Alpha Power ◽  
Heavy Tailed

Heavy tailed distributions have a big role in studying risk data sets. Statisticians in many cases search and try to find new or relatively new statistical models to fit data sets in different fields. This article introduced a relatively new heavy-tailed statistical model by using alpha power transformation and exponentiated log-logistic distribution which called alpha power exponentiated log-logistic distribution. Its statistical properties were derived mathematically such as moments, moment generating function, quantile function, entropy, inequality curves and order statistics. Five estimation methods were introduced mathematically and the behaviour of the proposed model parameters was checked by randomly generated data sets and these estimation methods. Also, some actuarial measures were deduced mathematically such as value at risk, tail value at risk, tail variance and tail variance premium. Numerical values for these measures were performed and proved that the proposed distribution has a heavier tail than others compared models. Finally, three real data sets from different fields were used to show how these proposed models fitting these data sets than other many wells known and related models.

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

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1276 ◽  
Author(s):  
Ahmed Z. Afify ◽  
Ahmed M. Gemeay ◽  
Noor Akma Ibrahim
Keyword(s):  
At Risk ◽  
Interest Rates ◽  
Exponential Distribution ◽  
Value At Risk ◽  
Computational Study ◽  
Risk Indicators ◽  
Alpha Power ◽  
Data Set ◽  
Heavy Tailed ◽  
Insurance Data

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.

scholarly journals A New Class of Heavy-Tailed Distributions: Modeling and Simulating Actuarial Measures

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Jin Zhao ◽  
Zubair Ahmad ◽  
Eisa Mahmoudi ◽  
E. H. Hafez ◽  
Marwa M. Mohie El-Din
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Weibull Model ◽  
Financial Data ◽  
Statistical Distributions ◽  
New Family ◽  
Heavy Tailed Distributions ◽  
Beta Power ◽  
Heavy Tailed ◽  
Modeling Data

Statistical distributions play a prominent role for modeling data in applied fields, particularly in actuarial, financial sciences, and risk management fields. Among the statistical distributions, the heavy-tailed distributions have proven the best choice to use for modeling heavy-tailed financial data. The actuaries are often in search of such types of distributions to provide the best description of the actuarial and financial data. This study presents a new power transformation to introduce a new family of heavy-tailed distributions useful for modeling heavy-tailed financial data. A submodel, namely, heavy-tailed beta-power transformed Weibull model is considered to demonstrate the adequacy of the proposed method. Some actuarial measures such as value at risk, tail value at risk, tail variance, and tail variance premium are calculated. A brief simulation study based on these measures is provided. Finally, an application to the insurance loss dataset is analyzed, which revealed that the proposed distribution is a superior model among the competitors and could potentially be very adequate in describing and modeling actuarial and financial data.

scholarly journals The Weibull Birnbaum-Saunders Distribution And Its Applications

2020 ◽  
Vol 9 (1) ◽  
pp. 61-81
Author(s):  
Lazhar BENKHELIFA
Keyword(s):  
Maximum Likelihood ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Reliability Estimation ◽  
Maximum Likelihood Estimates ◽  
Model Parameters ◽  
Data Sets ◽  
Proposed Model ◽  
Modeling Data

A new lifetime model, with four positive parameters, called the Weibull Birnbaum-Saunders distribution is proposed. The proposed model extends the Birnbaum-Saunders distribution and provides great flexibility in modeling data in practice. Some mathematical properties of the new distribution are obtained including expansions for the cumulative and density functions, moments, generating function, mean deviations, order statistics and reliability. Estimation of the model parameters is carried out by the maximum likelihood estimation method. A simulation study is presented to show the performance of the maximum likelihood estimates of the model parameters. The flexibility of the new model is examined by applying it to two real data sets.

Efficient Computation of Value at Risk with Heavy-Tailed Risk Factors

2009 ◽  
Author(s):  
Cheng-der Fuh ◽  
Inchi Hu ◽  
Kate Hsu ◽  
Ren-Her Wang
Keyword(s):  
Risk Factors ◽  
At Risk ◽  
Value At Risk ◽  
Efficient Computation ◽  
Heavy Tailed

scholarly journals Estimation and application in log-Fréchet regression model using censored data

2017 ◽  
Vol 5 (1) ◽  
pp. 23 ◽  
Author(s):  
Hanan Alamoudi ◽  
Salwa‎ Mousa‎ ◽  
Lamya Baharith
Keyword(s):  
Maximum Likelihood ◽  
Regression Model ◽  
Censored Data ◽  
Empirical Distribution ◽  
Model Parameters ◽  
Monte Carlo Simulation Study ◽  
Proposed Model ◽  
Data Validity ◽  
Global Influence ◽  
Deviance Residuals

This article introduces a new location-scale regression model based on a log-Fréchet distribution. Maximum likelihood and Jackknife methods are used to estimate the new model parameters for censored data. Martingale and deviance residuals are obtained to check model assumptions, data validity, and detect outliers. Moreover, global influence is used to detect influential observations. Monte Carlo simulation study is provided to compare the performance of the maximum likelihood and jackknife estimators for different sample sizes and censoring percentages. The empirical distribution of the martingale and deviance residuals of the proposed model is examined. A real lifetime heart transplant data is analyzed under the log-Fréchet regression model to illustrate the satisfactory results of the proposed model.

scholarly journals MULTIVARIATE HEAVY-TAILED MODELS FOR VALUE-AT-RISK ESTIMATION

2012 ◽  
Vol 15 (04) ◽  
pp. 1250029 ◽  
Author(s):  
CARLO MARINELLI ◽  
STEFANO D'ADDONA ◽  
SVETLOZAR T. RACHEV
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Stock Price ◽  
Risk Estimation ◽  
Price Data ◽  
Heavy Tailed Distributions ◽  
Derivative Instruments ◽  
Heavy Tailed ◽  
Student’S T

For purposes of Value-at-Risk estimation, we consider several multivariate families of heavy-tailed distributions, which can be seen as multidimensional versions of Paretian stable and Student's t distributions allowing different marginals to have different indices of tail thickness. After a discussion of relevant estimation and simulation issues, we conduct a backtesting study on a set of portfolios containing derivative instruments, using historical US stock price data.

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