The Arctan‐X Family of Distributions: Properties, Simulation, and Applications to Actuarial Sciences

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.

Radial Based Approximations for Arcsine, Arccosine, Arctangent and Applications

“…As an example, the relative error in approximations for asin(y) 6 , as defined by S 6,n (y) (Equation ( 130)) and the Borwein approximation S 6,n (y), are shown in Figure 15. The clear advantage of the root based approach over the series defined by S 6,n (y) is evident.…”

“…The Taylor series for arctangent, as given by Equation ( 7), leads to the 𝑛𝑡ℎ order approximation, 𝑇 , , for 𝑇: Graph of the relative error in approximations to asin(y) 6 , as defined by S 6,n (y) for n ∈ {3, 4, 5, 6, 7, 8, 9, 10}, along with root based approximations s 6,n (y) of orders 2, 3, 4, 5.…”

“…The arctangent function, for example, is found in the solution of the sine-Gordon partial differential equation for the case of soliton wave propagation, e.g., [5]. In statistical analysis the arcsine distribution is widely used and the arctangent function is the basis of a wide class of distributions, e.g., [6]. The graphs of sine, cosine, arcsine and arccosine are shown in Figure 1.…”

“…Figure15. Graph of the relative error in approximations to asin(y)6 , as defined by S 6,n (y) for n ∈ {3, 4, 5, 6, 7, 8, 9, 10}, along with root based approximations s 6,n (y) of orders 2, 3, 4, 5.…”

A Flexible Bayesian Parametric Proportional Hazard Model: Simulation and Applications to Right-Censored Healthcare Data

“…It has an additional parameter compared to the exponential. The additional parameter describes the shape of the hazard functions, based on the value of the shape parameter [ 21 ]. The pdf, cdf, sf, hrf, and chf of the Weibull random variable are, respectively, as follows.…”