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 E-mail: [email protected] ISSN: 2580-278X (e) 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
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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) fnrN ( ; ppp ,)(1) 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 characterization of infinite divisibility from a rp(1) , (4) VarNN () compound negative binomial distribution where p2 the summed random variables have an r p (5) exponential distribution. MtN ( ). 1 (1)exp(pt ) 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
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Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility …
where t (,) . The form e x p (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 (t) . (11) r X it p (7) N ().t 1(1)exp[]pit 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 X X12, X,. . . , 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 fxx()exp(), 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 propositions give some refinement properties of 1 compound negative binomial distribution related E(X ) , (8) to uniform continuity and positively defined 1 function. VarX() , (9) 2
p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.9 Figure 1. Parametric curves of negative binomial distribution’s characteristic function with various parameters p and r 2,4,6,8,10,12 for each graph.
0.4
0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
Figure 2. Parametric curves of exponential distribution’s characteristic function for various parameters λ = 0.1, 0.5, 1, 2, 4, 8.
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Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility …
Proposition 1. Suppose that S is a generated moment generating function and characteristic random variable from the sum of exponential function are defined as, respectively: r distribution as the compound negative binomial p pt distribution and has parameters (r, p,) . Its M (t) , (16) S p t expectation and variance are defined as. r r (1 p) E(S) , (12) p p S (t) . (17) r (1 p)(1 p) Var(S) . (13) 1 (1 p) (p)2 it Proof. The definition of this generated random Proof. By applying the definition of compound variable and the linearity property of expectation distribution’s moment generating function and are used to get the expectation of this compound the linearity property of its expectation, we get distribution, it is. for independent and identically random variable S as follows. NN ESEXEEXN() N ii MtEMtMMtSXNX()(())(ln(())) ii11 r (18) rp(1) ppt ENEXENEX( ())()(). . p pt (14) It is used moment generating function from From the expectation of negative binomial negative binomial and exponential distribution distribution in (3) and expectation of exponential in (5) and (10) then we have a proof for equation distribution in (8), we have a proof for equation (16). While by using the expectation’s linearity (12). While by using the definition of variance property, then we get and linearity of expectation, then it is obtained N SXNX(tEtMt )(( ))(ln(( ))) N N r Var(S) E Var X i N Var E X i N (19) i1 i1 p . r (1 p)(1 p) E(N)Var(X ) Var(N)(E(X ))2 . 1(1)p (p)2 it It is used the negative binomial distribution’s (15) moment generating function in (5) and By taking expectation from negative binomial exponential distribution’s characteristic function distribution and exponential distribution in (3) in (11), so equation (17) is proven. ■ and (8) also its variance in (4) and (9), we have a proof for equation (13).■ The characteristic function's characterization Proposition 2. Suppose that S is a generated from this compound negative binomial random variable from the sum of exponential distribution for various parameters is shown by distribution as the compound negative binomial parametric curves in Figure 3, Figure 4, Figure distribution and has parameters . Its 5, and Figure 6.
1.0 1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5 0.5
1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0
0.5 0.5 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.9 Figure 3. Characteristic function’s parametric curves from compound distribution as negative binomial with the sum of an exponential distribution for various parameters p where r 2, 4,6,8,10and λ = 0.5 in each graph.
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Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility …
1.0 1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5 0.5
1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0
0.5 0.5 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 r = 22 r = 23 r = 24 r = 25 r = 26 Figure 4. Characteristic function’s parametric curves from compound distribution as negative binomial with the sum of an exponential distribution for various parameters r where p 0.1,0.3,0.5,0.7,0.9 and λ = 0.5 in each graph.
1.0 1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5 0.5
1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0
0.5 0.5 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.9 Figure 5. Characteristic function’s parametric curves from compound distribution as negative binomial with the sum of an exponential distribution for various parameters p where 2, 4,8,16,32and r = 10 in each graph.
1.0 1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5 0.5
1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0
0.5 0.5 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 r = 2 r = 4 r = 6 r = 8 r = 10 Figure 6. Characteristic function’s parametric curves from compound distribution as negative binomial with the sum of an exponential distribution for various parameters r where 2,4,8,16,32 and p = 0.5 in each graph.
The characteristic function from a random The next propositions will tell the major variable that follows the compound distribution result of characteristic function properties from described in (1), as shown in the parametric a compound negative binomial distribution with curve above, is uniformly continuous. Its some of exponential distribution. A characteristic function tends to one for various characteristic function from a distribution is a parameters λ and r when parameter p tends to function with complex values defined in one, such as in Figure 3 and Figure 5. The greater Bochner’s theorem, as explained by (Kendall & parameter r then more complex graphs of Stuart, 1961). The definition says that a function parametric curves such as in Figure 4, while the (t) that maps the real number t to the complex shape of the parametric curve does not change value is called characteristic function if in any for various parameter λ as in Figure 6. This case, ()t is equal to 1 whent 0 and ()t is a parametric curve’s shape never disappears from non-negative definite function. the complex-plane and always depends on the The definition of non-negative definite parameter (r, p,) . It is more complicated than function is when it satisfies two terms that are a the representation of characteristic function from continuous function and the sum the negative binomial distribution or exponential is real and non-negative c j clS (t j tl ) 0 distribution. 1 jn1ln
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Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility …
for any number n which is positive integer, any (1p ) E (exp( i ( tjl t ) X )) for p (0,1) and real number t1,t2,...,tn and any complex number EittX(exp(())1jl, we have c1,c2,...,cn,. ccttjlSjl () 1,1j lnln r Proposition 3. The characteristic S ()t contains (r, p,) p the parameter as written in (17) from a cc jl random variable S as compound negative 11jnln 1(1)p i() tt binomial distribution with the sum of jl r p exponential distribution satisfies cc jl (i) (0) 1; 11jnln 1(1)(exp( pEi ())) ttX jl S mr r (ii) S (t) is uniformly continuous; pccpEi ttX jljl (1)(exp( ()). 110jnlnm (iii) is a positively defined function with (23) quadratic form Using the conjugate properties in the complex (20) ccttjlXjl ()0 number, so we gain the following equation. 11jnln cj ctt l Sjl() for real number t1,t2 ,. . . , tn and any 11j nl n mr complex number c1 ,c2 ,. . . , cn . r pcp Eitjj X(1)(exp() 10j nm Proof. (i) It is easy to obtain S (0) 1. mr (ii) The uniform continuity of is gained cpll Eit(1)(exp() X 10l nm by using the definition of continuity. The 2 mr definition says that for any 0 , always there r pcp Eitjj X (1)(exp() is a 0 so for tt12 then SS()().tt12 10j nm The value of only depends on . Therefore, mr 2 we get : r pcpj (1)0. S (t1) S (t2 ) 10j nm itj r r 1 1 (24) pr . (t) 1 (1 p)X (t1) 1 (1 p)X (t2 ) So S is proved as a positively defined function with the quadratic form as written in (21) (20) that has non-negative values.■ Then suppose h t1 t2 , so for h 0, the We can conclude that is a limit of equation (21) is approaching zero like characteristic function from random variable S written below: which is generated from the sum of exponential rr 11 distribution and follows the compound negative 0. 1(1)()1(1)(phtptXX ) 22binomial distribution where the parameters are (r, p,). (22) Whent 0, its value is equal to 1. It is a positively defined function that never disappears This equation holds for and tt 12 from the complex plane, such as proven in where S (h t2) S (t2) . Then is Proposition 3 and shown in the parametric curve uniformly continuous. at Figures 3-6.
Proposition 4. The function (iii) We will prove that the characteristic r functionS (t) is a positively defined function n that has quadratic form written in equation (20). p (t) (25) Using the characteristic function in equation (17) Sn 1 (1 p) and geometric series where the ratio is it is a characteristic function.
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Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility …
Proof. Based on Proposition 3, then it satisfies REFERENCES the characteristic function’s criteria of Bochner’s Artikis, T. (1983). Constructing infinitely Theorem. It is easy to say that the function (t) Sn divisible characteristic functions. is a characteristic function from random variable Archivum Mathematicum, 19(2), 57–61. Sn, which is gained from the sum of exponential Chen, X., & Guisong, C. (2017). Exact distribution, and follows the compound negative distribution of the convolution of negative binomial distribution, which has parameters binomial random variables. (r / n, p,). Communications in Statistics - Theory Proposition 5. The sum of exponential and Methods, 46(6), 2851–2856. distribution which construct the new compound https://doi.org/10.1080/03610926.2015.1 negative binomial distribution is infinitely 053931 divisible. Devianto, D. (2016). The uniform continuity of Proof. We use it the definition of infinite characteristic function from convoluted divisibility in (Artikis, 1983) to prove this part. exponential distribution with stabilizer It will show that for every n, which is positive constant. AIP Conference Proceedings, integer number, there exist Sn (t) , a 1707. https://doi.org/10.1063/1.4940863 characteristic function that satisfies Devianto, D., Oktasari, L., & Anas, M. (2015). Convolution of Generated Random (t) ( (t))n . Based on Proposition 4, we S SN Variable from Exponential Distribution have Sn (t) in equation (25) so that with Stabilizer Constant. Applied r Mathematical Sciences, 9(96), 4781– 4789. n p (S (t)) S (t) https://doi.org/10.12988/ams.2015.55396 n 1 (1 p) Devianto, D., Sarah, H., Yozza, F., & Yanuar, it M. (2019). The Infinitely Divisible (26) Characteristic Function of Compound We obtain (t) as a characteristic function S Poisson Distribution as the Sum of from compound negative binomial distribution Variational Cauchy Distribution. Journal with the sum of exponential distribution which of Physics: Conference Series, 1397, has parameters (r, p,). Because of its fulfillment 12065. https://doi.org/10.1088/1742- n 6596/1397/1/012065 the condition of (t) (t) for each t, so that S Sn Furman, E. (2007). On the convolution of the the characteristic function is infinitely negative binomial random variables. divisible.■ Statistics and Probability Letter, 77, 169– 172. Girondot, M. (2017). Optimizing sampling CONCLUSION design to infer the number of marine This paper introduces the new compound turtles nesting on low and high density sea negative binomial distribution, which is turtle rookeries using convolution of generated from the sum of exponential negative binomial distribution. Ecological distribution. Its characterization is gained by Indicators, 81, 83–89. applying and combining some properties of https://doi.org/10.1016/j.ecolind.2017.05. negative binomial distribution and exponential 063 distribution. It is using mathematical analysis Kendall, M. G., & Stuart, A. (1961). The methods to explore its characteristic function Advanced Theory of Statistics. Third and show uniform continuity property. The plot Edition. Hafner Publishing Company. of its characteristic function as a parametric Lukacs, E. (1992). Characteristic Function. curve never disappears from the complex plane. Hafner Publishing Company. It is also a positively defined function. This Omair, M., Almuhayfith, F., & Alzaid, A. compound’s characteristic function satisfies the (2018). A bivariate model based on definition of an infinitely divisible distribution. compound negative binomial distribution. Revista Colombiana de Estadística, 41(1),
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Anis Nur Afifah, et.al. The Characterization of Infinite Divisibility …
87–108. /2020/9398309/ https://doi.org/10.15446/rce.v41n1.57803 Sato, K. (2013). Levy Processes and Infinitely Ong, S. H., Toh, K., & Low, Y. C. (2019). The Divisible Distributions. Cambridge non-central negative binomial Studies in Advance Mathematics. distribution: Further properties and Wang, X., Zhao, X., & Sun, J. (2019). A applications. Communications in compound negative binomial distribution Statistics-Theory and Methods. with mutative termination conditions Panjer, H. H., & Willmot, G. E. (1981). Finite based on a change point. Journal of sum evaluation of the negative binomial- Computational and Applied Mathematics, exponential model. Astin Bulletin, 133– 351, 237–249. 137. Wang, Z. (2011). One mixed negative binomial Rono, A., Ogutu, C., & Weke, P. (2020). On distribution with application. Journal of Compound Distributions for Natural Statistical Planning and Inference, 141, Disaster Modelling in Kenya. 1153–1160. International Journal of Mathematics and https://www.sciencedirect.com/science/ar Mathematical Sciences, Volume 202. ticle/pii/S0378375810004313 https://www.hindawi.com/journals/ijmms
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