New methods to define heavy-tailed distributions with applications to insurance data

Abstract

Heavy-tailed distributions play an important role in modelling data in actuarial and financial sciences. In this article, nine new methods are suggested to define new distributions suitable for modelling data with an heavy right tail. For illustrative purposes, a special sub-model is considered in detail. Maximum likelihood estimators of the model parameters are obtained and a Monte Carlo simulation study is carried out to assess the behaviour of the estimators. Furthermore, some actuarial measures are calculated. A simulation study based on these actuarial measures is done. The usefulness of the proposed model is proved empirically by means of two real data sets. Finally, Bayesian analysis and performance of Gibbs sampling for the data sets are also carried out.

Comparison of COVID-19 Pandemic Dynamics in Asian Countries with Statistical Modeling

“…Note: Here, pm is used for proposed model. library(AdequacyModel) data=c(14, 9,8,5,20,1,3333,9,4,178,240,3993,69,72,93,6,7,1,4,19,65,3,2,60,182,6,41,6,200,7,1,2,5,30,812,12,3) sort(data) data ################# PDF of the proposed model pdf_pm <function(par, plot(x,ecdf,l-ty=1,lwd=4,type="s",xlab="x",ylab="G(x; 0.3147927, 0.2818076)", ylim=c(0,1),xlim=c(min(x),max(x)),col="black") par(new=TRUE) plot(x,proposedcdf,xlab="x",ylab="G(x; 0.3147927, 0.2818076)", ylim=c(0,1),xlim=c(min(x),max(x)),col="red",lty =5, lwd=4,type="l") par(new=TRUE) legend(1500, 0.4,c("Real Data","Proposed Model"),-col=c(1,2), lty =c(1,5), bty="n", cex=1.2) B.3. R Code for Plotting the Fitted Survival Function.…”

“…A model with lowest values for these statistics is considered a best candidate model. The formulae for these measures can be explored in [ 12 ]…”

“…with lowest values for these statistics is considered a best candidate model. The formulae for these measures can be explored in [12] For the COVID-19 total death data of the Asian countries, the estimates with the standard error (in parentheses) of the model parameters are provided in Table 3. The analytical measures of the NFE-Weibull and other considered models are provided in Tables 4 and 5.…”

“…) { theta=par[1] eta=par[2] theta * 2 * eta * (x^(theta-1)) * exp(-eta * x^theta) * ((1-exp(-eta * x^theta))) * (2-((1-exp(eta * x^theta))^2)) * (1/(exp((1-exp(-eta * x^theta))^2))) } ################# CDF of the proposed model cdf_pm <function(par,x) { theta=par[1] eta=par[2] 1-((1-((1-exp(-eta * x^theta))^2))/(exp((1-exp(-eta * x^theta))^2))) }set.seed(0) goodness.fit(pdf=pdf_pm, cdf=cdf_pm, starts = c(1,1), data = data, method="SANN", domain=c(0,Inf),mle=NULL) B.2. R Code for Plotting the Estimated Distribution Function.x=c(14,9,8,5,20,1,3333,9,4,178,240,3993,69,72,93,6,7,1,4, 19,65,3,2,60,182,6,41,6,200,7,1,2,5,30,812,12,3) sort(x) x #NAPTW theta=0.3147927 eta=0.2818076 x<-sort(x) F1<-ecdf(x) ecdf<-F1(c(x)) proposedcdf<-1-((1-((1-exp(-eta * x^theta))^2))/(exp((1-exp(-eta * x^theta))^2)))…”

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

“…Therefore, serious attempts have been made to propose new statistical models and are still growing rapidly. The new contribution is made via different approaches such as (i) transformation of variables, (ii) composition of two or more distributions, (iii) compounding of distributions, and (iv) finite mixture of distributions, see Ahmad et al [ 4 ].…”