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 dis… Show more
“…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.…”
Section: Resultsmentioning
confidence: 99%
“…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.…”
Section: Approximations For the Inverse Tangent Integral Functionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Based on the geometry of a radial function, a sequence of approximations for arcsine, arccosine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is significantly better than Taylor series approximations for arguments approaching one. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of 10−16, and lower, can be defined. Applications of the approximations include: first, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with significantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approximations to arccosine and arcsine, to even order powers, with relative errors that are significantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals.
“…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.…”
Section: Resultsmentioning
confidence: 99%
“…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.…”
Section: Approximations For the Inverse Tangent Integral Functionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Based on the geometry of a radial function, a sequence of approximations for arcsine, arccosine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is significantly better than Taylor series approximations for arguments approaching one. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of 10−16, and lower, can be defined. Applications of the approximations include: first, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with significantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approximations to arccosine and arcsine, to even order powers, with relative errors that are significantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals.
“…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.…”
Section: Distributions Closed Under Ph Frameworkmentioning
Survival analysis is a collection of statistical techniques which examine the time it takes for an event to occur, and it is one of the most important fields in biomedical sciences and other variety of scientific disciplines. Furthermore, the computational rapid advancements in recent decades have advocated the application of Bayesian techniques in this field, giving a powerful and flexible alternative to the classical inference. The aim of this study is to consider the Bayesian inference for the generalized log-logistic proportional hazard model with applications to right-censored healthcare data sets. We assume an independent gamma prior for the baseline hazard parameters and a normal prior is placed on the regression coefficients. We then obtain the exact form of the joint posterior distribution of the regression coefficients and distributional parameters. The Bayesian estimates of the parameters of the proposed model are obtained using the Markov chain Monte Carlo (McMC) simulation technique. All computations are performed in Bayesian analysis using Gibbs sampling (BUGS) syntax that can be run with Just Another Gibbs Sampling (JAGS) from the R software. A detailed simulation study was used to assess the performance of the proposed parametric proportional hazard model. Two real-survival data problems in the healthcare are analyzed for illustration of the proposed model and for model comparison. Furthermore, the convergence diagnostic tests are presented and analyzed. Finally, our research found that the proposed parametric proportional hazard model performs well and could be beneficial in analyzing various types of survival data.
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