Abstract
The purpose of this paper is to investigate a new family of distributions based on an inverse trigonometric function known as the arctangent function. In the context of actuarial science, heavy-tailed probability distributions are immensely beneficial and play an important role in modelling data sets. Actuaries are committed to finding for such distributions in order to get an excellent fit to complex economic and actuarial data sets. The current research takes a look at a popular method for generating new distributions which are excellent candidates for dealing with heavy-tailed data. The proposed family of distributions is known as the Arctan-X family of distributions and is introduced using an inverse trigonometric function. For the specific purpose of the show of strength, we studied the Arctan-Weibull distribution as a special case of the developed family. To estimate the parameters of the Arctan-Weibull distribution, the frequentist approach, i.e., maximum likelihood estimation, is used. A rigorous Monte Carlo simulation analysis is used to determine the efficiency of the obtained estimators. The Arctan-Weibull model is demonstrated using a real-world insurance data set. The Arctan-Weibull is compared to well-known two-, three-, and four-parameter competitors. Among the competing distributions are Weibull, Kappa, Burr-XII, and beta-Weibull. For model comparison, we used the most precise tests used to know whether the Arctan-Weibull distribution is more useful than competing models.
Highlights
Numerous disciplines of study have examined heavy-tailed probability distributions, including actuarial science, biomedical sciences, engineering, risk management, and economics
We explore the maximum likelihood estimation (MLE) method's performance in estimating the AT-W parameters using an MC simulation study with 750 repetitions
We determine the mean of the estimated parameters, the absolute bias, the mean square error (MSE), and the root mean square error (RMSE), and the following steps were followed: (i) We generated the samples by inverting the cdf given in [20]
Summary
Numerous disciplines of study have examined heavy-tailed probability distributions, including actuarial science, biomedical sciences, engineering, risk management, and economics. Some procedures have been proposed to generate a new class of heavy-tailed probability distributions with adequate description and a high degree of flexibility. Among these techniques, the use of trigonometric functions and their inverses has been at the forefront of the development of new families of probability distributions. One of the really essential functions of financial and actuarial science is the accurate forecasting of large monetary financial losses Underestimation of such losses exposes the company to serious operational risks, including such bankruptcy and underestimating premium. To mitigate such circumstances and provide precise forecasts of actuarial science losses, actuaries frequently propose flexible heavytailed distributions
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