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The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution As the Sum of Exponential Distribution

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The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution As the Sum of Exponential Distribution SAINSTEK: JURNAL SAINS DAN TEKNOLOGI June 2021 Vol 13 No 1 Publisher: AMSET IAIN Batusangkar and IAIN Batusangkar Press ISSN: 2085-8019 (p) Website: http://ecampus.iainbatusangkar.ac.id/ojs/index.php/sainstek ISSN: 2580-278X (e) E-mail: [email protected] pp : 58-65 The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Exponential Distribution Anis Nur Afifah1, Maiyastri1, Dodi Devianto1* 1Department of Mathematics, Faculty of Mathematics and Natural Sciences, Andalas University, Padang 25163, West Sumatra Province, INDONESIA *Email: [email protected] Article History Abstract Received: 8 January 2021 The sum of random variables that are identical and independent Revised: 9 February 2021 from an exponential distribution creates the compound distribution. Accepted: 6 June 2021 It is called compound negative binomial distribution as the sum of Published: 30 June 2021 exponential distribution when the number of random variables added follows the negative binomial distribution. This compound Key Words distribution’s characteristic function is established by using Compound distribution; mathematical analysis methods, included its uniform continuity Characteristic function; property. The characteristic function's parametric curves never Infinitely divisible disappear from the complex plane, which means it is a positively distributions; defined function. Another characteristic function's property shows Negative binomial that this compound distribution is one of infinitely divisible distribution; distribution. Exponential distribution. INTRODUCTION The negative binomial-exponential model, as introduced by (Panjer & Willmot, 1981) is one The sum of random variables that are of compound negative binomial distribution identical and independent where the number of where its convolution is evaluated as finite sums. random variables added follows a negative Furthermore, (Z. Wang, 2011) applied one binomial distribution forms as the compound mixed negative binomial distribution in the negative binomial distribution. These identical insurance. We have found another application of and independent random variables added follow compound negative binomial distribution in an exponential distribution, it is called many fields, such as it has done by (Rono et al., compound negative binomial distribution with 2020) in modeling natural disaster in Kenya. the sum of exponential distribution. For an (Girondot, 2017) used the convolution of instant, suppose the random variable N has negative binomial distribution to optimize the negative binomial distribution and X , X ,…, X 1 2 N turtle research sampling design. (Omair et al., are identical and independent random variables 2018) carried another application out by utilizing from an exponential distribution. We denote the this distribution to develop a new bivariate sum of these random variables by model that is suitable for accident data. SN X1 X 2 ... X N (1) This compound distribution has better as the compound distribution. theoretical development from statistical (Furman, 2007), who delivered the mathematics views, as explained by previous convolution of some independent random authors and its applications. (Chen & Guisong, variables as the sum of negative binomial 2017) derived the exact distribution using the distribution and its properties, introduced the generating function and the convolution, while previous research on compound distribution. (X. Wang et al., 2019) studied the characters of Sainstek: Jurnal Sains dan Teknologi 58 Vol 13 No 1, June 2021 ISSN: 2085-8019 (p), ISSN: 2580-278x (e) Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility … compound negative binomial distribution got properties of the two constituent distributions, from the finite Markov chain approach. namely the negative binomial distribution and Furthermore, (Ong et al., 2019) described the the exponential distribution. It is determined the various important properties like unmodelled, sum of exponential distribution formed the log-concavity, and asymptotic behavior from the compound negative binomial distribution then compound negative binomial distribution and its properties are derived by using mathematical their use in practical applications. analysis based on its characteristic function. The application of compound distribution Parametric curves were used as a tool to show as the convolution where the random variables the properties of the characteristic functions. We added has an exponential distribution is widely examined it using Bochner’s theorem to show used in insurance. We often assume the that a function satisfies the condition and aggregate severity loss as an exponential sufficient requirements as a characteristic distribution. Theoretically, (Devianto et al., function. The definition of the infinite 2015) proposed new distribution from the divisibility of a distribution based on its convolution of random variables that has an characteristic function is used in the deductive exponential distribution with stabilizer constant. method to prove that it is an infinitely divisible They also suggested some essential characters distribution. from hypoexponential distribution with stabilizer constant. Moreover, (Devianto, 2016) explained the characteristic function's uniform RESULTS AND DISCUSSION continuity properties from the previously The density probability function of proposed distribution. (Devianto et al., 2019) negative binomial distribution with parameter discussed another study of the uniform (r,p) for random variable N is defined as continuity of other compound distribution's rn1 characteristic functions about Poisson rn (2) fN ( n ; r , p ) p (1 p ) compound distribution where the random n variables added has Cauchy variational where n 0,1, 2,... r 0 , and p (0,1) . The distribution. value of n represents the number of failures It confirms that characteristic function has before a certain number of successes in the essential roles in the convolution and the experiment. This specific number of successes compound distribution. The characterization of denoted as r. We expressed the probability of compound distribution with convolution from success in an individual experiment as p, while several special distributions is an important the probability of failure is stated as q 1 p . property to establish with its refinement in Its expected value, variance, and moment application. The leading theory in determining generating function are written as follows, the properties of the characteristic function was respectively: proposed by (Lukacs, 1992) and developed by rp(1 ) , (3) (Artikis, 1983) to construct an infinitely divisible ENN () distribution. This paper presents the p rp(1 ) characterization of infinite divisibility from a , (4) VarN () N compound negative binomial distribution where p2 r the summed random variables have an p (5) exponential distribution. MtN (). 1 (1pt )exp( ) The other negative binomial distribution’s METHOD properties can be determined by characteristic function. (Lukacs, 1992) and (Sato, 2013) have The research method in this paper is a noted the characteristic function’s definition literature study on the infinite divisibility using the Fourier-Stieltjes transform. We wrote property and compound negative binomial the characteristic function of random variable X distribution’s characterization gained from the as sum of exponential distribution. First, we (t ) E [exp( itX )] (6) defined the characteristic function and other X Sainstek: Jurnal Sains dan Teknologi 59 Vol 13 No 1, June 2021 ISSN: 2085-8019 (p), ISSN: 2580-278x (e) Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility … where t (,) . The form exp(itX ) can be written as cos tX + i sin tX where i 1 . So M X (t) , (10) that the characteristic function for random t variable N as negative binomial distribution is r X (t) . (11) p it (7) N ().t 1 (1p )exp[ it ] The parametric curves of the exponential The shape of parametric curves shows the distribution’s characteristic function with characterization of a characteristic function. The various parameters λ are shown in Figure 2. It is parametric curves are got by plotting the uniformly continuous and is a circle with a characteristic function at the Cartesian radius of 0.5 for all of various λ. coordinates system. The real part of the Suppose that random variable N has characteristic function is represented as the x- negative binomial distribution with parameter axis, while we illustrate its imaginary part as the (r,p). Let XXX12, ,..., N be identical and y-axis. For the negative binomial distribution, independent random variables from an we plot it in Figure 1. Its characteristic function exponential distribution with parameter λ. These is uniformly continuous and for p tends to one, summed random variables construct a new the characteristic function tends to one. random variable that has negative binomial We defined the exponential distribution compound distribution with the sum of from a random variable X as a continuous exponential distribution SN X1 X2 ... X N probability distribution function with parameter contains the parameter (r,p,λ). This distribution's λ as f( x ) exp( x ), where 0 and characterization is a critical theory to determine x 0 . The expected value, variance, moment since it’s widely used in insurance modeling for generating function, and its characteristic the severity lost model. The following function are, respectively
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