Hindawi Complexity Volume 2020, Article ID 8394815, 18 pages https://doi.org/10.1155/2020/8394815 Research Article The Arcsine Exponentiated-X Family: Validation and Insurance Application Wenjing He,1 Zubair Ahmad ,2 Ahmed Z. Afify,3 and Hafida Goual4 1College of Modern Economics and Management, Jiangxi University of Finance and Economics, Nanchang, Jiangxi, China 2Department of Statistics, Yazd University, P.O. Box 89175-741, Yazd, Iran 3Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt 4Laboratory of Probability and Statistics University of Badji Mokhtar, Annaba, Algeria Correspondence should be addressed to Zubair Ahmad; [email protected] Received 13 January 2020; Revised 26 March 2020; Accepted 7 April 2020; Published 14 May 2020 Academic Editor: Dimitri Volchenkov Copyright © 2020 Wenjing He et al. .is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. .ese actuarial measures are also estimated from insurance claims real data for the ASE-W and other competing distributions. .e 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. .e modified test shows that the ASE-W model can be used as a good candidate for analyzing heavy tailed insurance claims data. lim exp(− cx) − 1 x⟶∞ � 0; c > 0; (1) 1. Introduction 1 − F(x) Heavy tailed distributions play a significant role in modeling where F(x) is the cdf of a baseline distribution. More in- data in applied sciences, particularly in risk management, formation can be explored in Resnick [8] and Beirlant et al. banking, economics, financial, and actuarial sciences. [9]. However, the quality of the procedures primarily depends Dutta and Perry [10] performed an empirical analysis of upon the assumed probability model of the phenomenon loss distributions to estimate the risk via different ap- under consideration. Among the applied fields, the insur- proaches. .ey rejected the idea of using the exponential, ance datasets are usually positive [1], right-skewed [2], gamma, and Weibull models because of their poor results unimodal shaped [3], and with heavy tails [4]. Right-skewed and concluded that one would need to use a model that is data may be adequately modeled by the skewed distributions flexible enough in its structure. .ese results encouraged the [5]. .erefore, a number of unimodal positively skewed researchers to propose new flexible models providing greater parametric distributions have been employed to model such accuracy in data fitting. .erefore, a number of approaches datasets [6, 7]. have been proposed to obtain new distributions with heavier .e heavy tailed distributions are those whose right tail tails than the exponential distribution, such as (i) trans- probabilities are heavier than the exponential one, that is, formation method [11, 12], (ii) composition of two or more 2 Complexity distributions [13], (iii) compounding of distributions 2 G(x; a; ξ) � arcsineF(x; ξ)a �; a > 0; ξ ∈ R; x ∈ R; [14, 15] , and (iv) finite mixture of distributions [16, 17]. π .e abovementioned approaches are very useful in de- (4) riving new flexible distributions; however, they are still subject to some sort of deficiencies, for example, (i) the where F(x; ξ) is the baseline cdf with a parameter vector ξ transformation approach is simple to apply, but its infer- and an additional shape parameter a. ences become difficult and many computational work is .e probability density function (pdf) corresponding to required to derive the distributional characteristics [18]. (ii) equation (4) is given by .e approach of composition of two or more distributions 2 af(x; ξ)F(x; ξ)a− 1 using a fixed or a priori known mixing weights, and hence g(x; a; ξ) � q����������� ; a > 0; ξ ∈ R; x ∈ R: π 2a they can be very restrictive [19]. To overcome this problem, 1 − F(x; ξ) Scollnik [20] used unrestricted mixing weights. (iii) .e (5) density obtained by the compounding approach may not always have a closed form expression which makes the .e new pdf is most tractable when F(x; ξ) and f(x; ξ) estimation more cumbersome [21]. (iv) Finite mixture have simple analytical expressions. Henceforth, a random models represent a further approach to define very flexible variable X with pdf equation (5) is denoted by distributions which are also able to capture, for instance, X ∼ ASE − X(x; a; ξ). Moreover, the key motivations for multimodality of the underlying distribution. .e price to using the ASE-X family in practice are the following: pay for this greater flexibility is a more complicated and (i) To improve the characteristics and flexibility of the computationally challenging inference [22]. existing distributions, the special models of this To overcome the problems associated with the above family can provide left-skewed, right-skewed, uni- former methods, many authors have proposed new families modal, reversed J-shaped and symmetric densities, of distributions, see, for example, Al-Mofleh [23], Jamal and and decreasing and increasing, bathtub, upside Nasir [24] and Nasir et al. [25], Ahmad et al. [26], Afify et al. down bathtub, and reversed-J hazard rates (See [27], Cordeiro et al. [28], Ahmad et al. [29], Afify and Figures 1 and 2) Alizadeh [30], and among many others. .erefore, bringing flexibility to the existing distributions by adding additional (ii) A very simple and convenient method of adding an parameter(s) is a desirable feature and an interesting re- additional parameter provide extended heavy tailed search topic. distributions which are very useful in modeling data In this regard, Mudholkar and Srivastava [31] intro- form the insurance field (see Sections 6 and 7) duced the exponentiated family of distributions by adding a (iii) To introduce the extended version of a baseline shape parameter to obtain more flexible version of the distribution with closed forms for the cdf and existing distributions. A random variable X is said to follow hazard rate function (hrf), the special submodels of the exponentiated family, if its cumulative distribution this family can be used in analyzing censored function (cdf) is given by datasets G(x; a; ξ) � F(x; ξ)a; a > 0; ξ ∈ R; x ∈ R; (2) (iv) .e special cases of the ASE-X approach is capable of modeling heavy tailed datasets in actuarial sci- where F(x; ξ) is the cdf of the baseline distribution ence as compared with existing competing models depending on the parameter vector ξ and a > 0 is an ad- (see Sections 6 and 7). ditional shape parameter. Using equation (2), the expo- nentiated versions of the existing distributions have been Using the new cdf in equation (4), a number of new proposed in the literature. flexible distributions can be obtained. Some new con- Furthermore, Cordeiro and de Castro [32] proposed tributed models based on the ASE-X approach are pre- another approach known as the Kumaraswamy-generalized sented in Table 1. (Ku-G) family by adding two additional shape parameters. .e survival function (sf) and hrf of the proposed family .e cdf of the Ku-G family is are, respectively, given by 2 a a b S(x; a; ξ) � 1 − arcsine F(x; ξ) �; a > 0; ξ ∈ R; x ∈ R; G(x; a; b; ξ) � 1 − �1 − F(x; ξ) � ; a; b; > 0; ξ ∈ R; x ∈ R: π (3) af(x; )F(x; )a− 1 ξ ξq����������� From equation (3), it is clear that, for b � 1, the Ku-G h(x; a; ξ) � ; ( ) − e() F(x; )a � − F(x; )2a � family reduces to the exponentiated family. For a contrib- π/2 arcsin ξ 1 ξ uted work based on equation (3), we refer to Ahmad et al. [33], Mead and Afify [34], Afify et al. [35], and Mansour et al. a > 0; ξ ∈ R; x ∈ R: [36]. (6) In this paper, we enrich the branch of distribution theory by introducing the heavy tailed arcsine exponentiated-X .e paper is outlined as follows. In Section 2, we define (ASE-X) family of distributions. A random variable X be- the ASE-W distribution and present some plots for its longs to the proposed ASE-X family if its cdf is density and hazard functions. We provide some Complexity 3 1.5 1.5 1.0 1.0 ) ) x x ( ( g g 0.5 0.5 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x x α = 0.5, a = 7.0, γ = 3 α = 3.1, a = 4.0, γ = 2 α = 1.8, a = 6.0, γ = 3 α = 3.7, a = 2.9, γ = 0.1 α = 1.2, a = 6.0, γ = 2 α = 2.9, a = 3.0, γ = 0.5 α = 1.5, a = 1.6, γ = 0.8 α = 3.2, a = 1.3, γ = 0.3 α = 0.8, a = 0.3, γ = 1.5 α = 3.8, a = 0.2, γ = 0.5 (a) (b) Figure 1: Different plots for the pdf of the ASE-W distribution for selected values of its parameters.