Approximations to the distribution of sum of independent non-identically gamma random variables

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1 Math Sci (205) 9: DOI 0.007/s ORIGINAL RESEARCH Approximations to the distribution of sum of independent non-identically gamma random variables H. Murakami Received: 3 May 205 / Accepted: 22 October 205 / Published online: 0 November 205 The Author(s) 205. This article is published with open access at Springerlink.com Abstract Calculating the sum of independent non-identically distributed random variables is necessary in the scientific field. Computing the probability of the corresponding significance point is important in ces that have a finite sum of random variables. However, it is difficult to evaluate this probability when the number of random variables increes. Under these circumstances, consideration of a more accurate approximation of the distribution function is extremely important. A saddlepoint approximation is performed using upper probabilities from the distribution of the sum of independent non-identically gamma random variables under finite sample sizes. In this study, we compared the results from a saddlepoint approximation to those from normal and moment-bed approximations to identify the most appropriate method to use for the distribution function. Keywords Independent and non-identically distributed Saddlepoint approximation Sum of gamma random variables Introduction The distribution of the sum of independent identically distributed gamma random variables is well known. However, within the scientific field, it is necessary to know the distribution of the sum of independent non-identically distributed (i.n.i.d.) gamma random variables. For example, it & H. Murakami murakami@gug.math.chuo-u.ac.jp Department of Mathematical Information Science, Tokyo University of Science, Tokyo, Japan would be necessary to know this distribution for calculating total waiting times component times are sumed to be independent exponential or gamma random variables. In addition, engineers calculate total excess water flow into a dam the sum of i.n.i.d. gamma random variables. To calculate the exact probability distribution of the sum of i.n.i.d. gamma random variables, the probability of all possible elements consistent with the sum must be computed. Mathai [2] derived the distribution of the sum of i.n.i.d. gamma random variables by converting the moment-generating function. Additionally, Moschopoulos [3] calculated the distribution of the sum of i.n.i.d. gamma random variables using a simple recursive relation approach. For the detail of the gamma distribution family, we refer the reader to Khodabin and Ahmadabadi [9]. However, Mathai [2] and Moschopoulos [3] derived the density of the sum of i.n.i.d. gamma random variables with infinite summation. This method of computation is intractable in practice, especially in ces in which there is an incree in the number of random variables. An exact calculation is feible by applying the standard inversion formula to the characteristic function in computer algebra systems, such Mathematica. However, in these calculations, the probability is estimated with an approximation method. Approximation methods are widely used and have been studied extensively. From a practical view, approximations are typically precise and straightforward to implement in various statistical software programs. Hence, obtaining a more accurate approximation for evaluating the density or the distribution function of i.n.i.d. random variables remains an important area of debate in statistics. In this study, we describe the use of approximation methods to calculate the distribution of the sum of i.n.i.d. gamma random variables in Sect. A Saddlepoint approximation to the distribution of sum of i.n.i.d. gamma random variables. Furthermore, we 23

2 206 Math Sci (205) 9: discuss the derivation of the order of errors of suggested approximation for the given distribution. For the approximation presented in this paper, we used the saddlepoint formula employed previously by Daniels [2, 3] and developed by Lugannani []. The saddlepoint approximation can be obtained for any statistic or random variable that contains a cumulant generating function. Additionally, the saddlepoint generates accurate probabilities in the tail of distribution. Saddlepoint approximations have been used with great success by several researchers. Excellent discussions of their applications to a range of distributional problems are found in the following studies: Jensen [8], Huzurbazar [7], Kolsa [0], and Butler []. Recently, Eisinga et al. [4] discussed the use of the saddlepoint approximation for the sum of i.n.i.d. binomial random variables. Additionally, Murakami [4] and Nadarajah [5] considered the use of the saddlepoint approximation for the sum of i.n.i.d. uniform and beta random variables, respectively. In Sect. Numerical results, we discuss the results obtained from using the saddlepoint approximation. In Sect. Concluding remarks, we summarize our conclusions. A Saddlepoint approximation to the distribution of sum of i.n.i.d. gamma random variables In this section, we discuss the use of the saddlepoint approximation of the sum of independent non-identically gamma random variables. We sumed that X... X n are independent random variables, with shape, a i [ 0, and scale parameters, b i [ 0, for i ¼... n. Next, we let S n ¼ X þ X 2 þþx n. The moment-generating function of S n is M n ðsþ ¼ Yn ð b i sþ a i : i¼ It is important to note that Mathai [2] derived the density function of the sum of i.n.i.d. gamma random variables by converting its moment-generating function follows: ( ) f Sn ðxþ ¼ Yn b ai i CðqÞ x q exp x i¼ X X n o ða 2 Þ r2 ða n Þ rn r 2¼0 r n¼0 r2 x rn x b b 2 b b n r 2! r n!ðqþ q x [ 0 b C f Sn ðxþ ¼ Cðq þ kþb qþk C ¼ Yn i¼ d kþ ¼ k þ x k ¼ k b ai b i X n i¼ X kþ i¼ X k¼0 d k x qþk exp x x [ 0 b ix i d kþi k ¼ a i ð b b i Þ k k ¼ 2... with d 0 ¼. It is difficult to evaluate the exact density of S n with increing n. Herein, we consider an approximation to the distribution of S n. The cumulant generating function of S n is j n ðsþ ¼ Xn a i logð b i sþ: i¼ Using the cumulant generating function, the mean, l, and variance, r 2,ofS n are given below: l ¼ Xn i¼ a i b i and r 2 ¼ Xn i¼ a i b 2 i : According to Daniels [3], the saddlepoint approximation of the density function of S n is follows: f s ðvþ ¼f ðvþf þ Oðn Þg f ðvþ ¼ 2pj 00 n ð^sþ 2 expfj n ð^sþ^svg j 0 nðsþ ¼Xn i¼ a i b i b i s j00 nðsþ ¼Xn i¼ a i b 2 i ð b i sþ 2 and ^s is the root of j 0 nðsþ ¼v which is readily solved numerically by the Newton Raphson algorithm. Several approaches have been used to further minimize the error of the saddlepoint approximation [5]. For example, one method uses a higher order approximation by including adjustments for the third and fourth cumulants [3]. A higher order saddlepoint approximation uses the following correction term: ( ) f s ðvþ ¼f ðvþ þ jn ð4þ ð^sþ 5 jn ð3þ ð^sþ 2 8 j 00 nð^sþ2 24 j þ 00 Oðn2 Þ n ð^sþ3 ðþ q ¼ a þþa n and ðyþ z denote the Pochhammer symbol. In addition, Moschopoulos [3] obtained the density function of the sum of i.n.i.d. gamma random variables using the following simple recursive relation approach: jn ð3þ ðsþ ¼Xn i¼ 2a i b 3 i ð b i sþ 3 jð4þ n ðsþ ¼Xn i¼ 6a i b 4 i ð b i sþ 4 : 23

3 Math Sci (205) 9: The approximate tail probabilities of S n are determined by numerically integrating Eq. (). An alternative approach is to use the Lugannani and Rice [] for the continuous tail probability approximation follows: PrðS n \vþ Uð ^wþ/ð^wþ ^u ^w /ðþ is the standard normal density function, UðÞ is the corresponding cumulative distribution function, and p ^w ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 2ð^sv j n ð^sþ Þsgnð^sÞ ^u ¼ ^s j 00 n ð^sþ sgnð^sþ ¼, 0 if ^s is positive, negative, or zero. Numerical results In this section, we investigated the upper probability using the saddlepoint approximations to S n. In this study, we focused on the Lugannani Rice formula. Note that Mathai [2] obtained a normal approximation with n!. Moschopoulos [3] derived the density of S n with infinite summation follows: C X f Sn ðxþ ¼ Cðq þ kþb qþk d k x qþk exp x x [ 0: b k¼0 We used a finite number and truncated the infinite series to meet an acceptable precision published by Moschopoulos [3]. This equation is listed follows: C X f Sn ðxþ ¼ Cðq þ kþb qþk d k x qþk exp x b k¼0 ¼ C CðqÞb q x q exp x X d k x k b ðqþ k¼0 k b C CðqÞb q x q exp x X bx k b k! b ¼ C CðqÞb q x q exp k¼0 xð bþ b ðqþ k ¼ qðq þ Þðq þ k Þ, ðqþ 0 ¼ and b ¼ max 2 i n ð b =b i Þ. To bound the truncation error with the sum of the first þ, we used the following equation: EðwÞ ¼ Z w C CðqÞb q x q xð bþ exp dx b C X 0 Cðq þ kþb qþk d k x qþk exp x b k¼0 0 Z w dx: In addition, we used another approximation method for the distribution of S n, a moment-bed approximation proposed by Ha and Provost [6]. The distribution of S n is approximated by the polynomial adjusted ~f k ðvþ such that ~f k ðvþ ¼wðvÞ Xk ¼0 n v 0 n 0 0 mð0þ mðþ mðk Þ mðkþ n mðþ mð2þ mðkþ mðk þ Þ B. A ¼ B A mðkþ mðk þ Þ mð2k Þ mð2kþ n k 0 EðMÞ B. A EðM k Þ and m(k) and EðM k Þ denote the kth moment of the adjusted distribution wðvþ and the kth moment of S n, respectively. Herein, we consider the approximation adjusted with the skew-normal distribution follows: w ðvþ ¼ 2 r / v g U kðv Þ g rffiffiffi 2 ¼ l dg g 2 ¼ r 2 2d2 p p f ¼ EðM3 p Þ3lr2 l 3 r 3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d u k ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi jdj ¼SignðfÞt p jnj 2=3 d 2 2 jnj 2=3 þ 4p 2=3 2 n ¼ min ð0:99 jfjþ: Then, ( mðkþ ¼ dk dt k 2 exp t þ g2 t 2!) kgt U pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ k 2 : t¼0 Note that we obtained n 0 ¼ n ¼ n 2 ¼ 0 for k ¼ 2. Afterwards, the moment-bed approximation with skewnormal polynomial w follows: ~f 2 ðvþ ¼w ðvþ X2 ¼0 n v ¼ w ðvþð þ 0 þ 0Þ ¼w ðvþ: An important step for the proposed method is to determine the optimal degrees for the polynomials. We followed the selection rule, which is bed on the integrated squared differences between density approximations previously published by Ha and Provost [6]. For this study, the following notations were utilized: exact probability of S n, E P, ( proposed by Moschopoulos 23

4 208 Math Sci (205) 9: Table Numerical results for a % significance level for ce (n ¼ 5Þ v E P A L A N A M r.e. A L r.e. A N r.e. A M Ce A Ce B Ce C Ce D Ce E Ce F

5 Math Sci (205) 9: Table 2 Numerical results for a % significance level for ce 2 (n ¼ 0) v E P A L A N A M r.e. A L r.e. A N r.e. A M Ce A Ce B Ce C Ce D Ce E Ce F

6 20 Math Sci (205) 9: Table 3 Numerical results for a % significance level for ce 3 (n ¼ 5) v E P A L A N A M r.e. A L r.e. A N r.e. A M Ce A Ce B Ce C Ce D Ce E Ce F

7 Math Sci (205) 9: [3]) normal approximation, A N saddlepoint approximation with Lugannani Rice formula, A L moment-bed approximation with skew-normal polynomial, A M and the relative error of approximations, r.e. (Tables, 2, 3). We used different values for a~ ¼ða a 2... a n Þ and b~ ¼ðb b 2... b n Þ. These values were grouped into ces 3. Herein, we sumed that a i and b i for n ¼ 5 0 and 5 follows: Ce : a i and b i were simulated from Ce A: Uniform distribution with interval [0, 3] independently a i ¼ð:04022 :5249 2:9665 0:7756 :93264Þ b i ¼ð2: :6082 2:49735 :57684 :05720Þ Ce B: Poisson distribution with parameter k ¼ 0 independently a i ¼ð Þ b i ¼ð Þ Ce C: Lognormal distribution with parameters location l ¼ 0 and scale r ¼ 2 independently a i ¼ð0: : :98562 : :40726Þ b i ¼ð: :3988 0: :4009 0:8285Þ Ce D: Gamma distribution with parameters shape c ¼ 2:0 and scale n ¼ :5 independently a i ¼ð:96560 : :0026 4:2882 3:0364Þ b i ¼ð6: : :508 3: :3223Þ Ce E: Exponential distribution with parameter k ¼ 2:0 independently a i ¼ð0: : : : :2986Þ b i ¼ð0:020 0: :0969 0:3260 0:5249Þ Ce F: Binomial distribution with parameter N ¼ 20, p ¼ 0:3 independently a i ¼ð Þ b i ¼ð Þ Ce 2: a i and b i were simulated from Ce A: Uniform [0, 3] independently a i ¼ð0:2247 :4752 0:50906 : :7236 2:50722 : : :6593 2:06543Þ b i ¼ð2: :9297 2: : :9546 2:0277 2: : :0206 :98484Þ Ce B: Poisson distribution with parameter k ¼ 0 independently a i ¼ð Þ b i ¼ð Þ Ce C: Lognormal distribution with parameters location l ¼ 0 and scale r ¼ 2 independently a i ¼ð0:560 0:5703 0:8732 0: :5800 0: :0573 0: :5837 0:50578Þ b i ¼ð0: :0076 0: :0232 0:3829 0: :6248 7:7075 0:445 0:04548Þ Ce D: Gamma distribution with parameters shape c ¼ 2:0 and scale n ¼ :5 independently a i ¼ð2:796 :0074 : : :898 3: : :7498 2:54858 :52722Þ 23

8 22 Math Sci (205) 9: b i ¼ð4:544 3: :6604 3:88386 : : : :670 : :7920Þ Ce E: Exponential distribution with parameter k ¼ 2:0 independently a i ¼ð0: : : : : :03837 :3507 0:368 0:2429 0:2935Þ b i ¼ð: :5572 :820 0:2383 0: :3853 0: :6255 0: :2336Þ Ce F: Binomial distribution with parameter N ¼ 20, p ¼ 0:3 independently a i ¼ð Þ b i ¼ð Þ Ce 3: a i and b i were simulated from Ce A: Uniform distribution with interval [0, 3] independently a i ¼ð0: : :56 :2297 2:29383 :8906 : :3849 2: : :98 2: :79358 : :46924Þ b i ¼ð0:2882 :9593 2:4996 0: : :4 : :3692 2:767 0:77938 : :7339 0:963 0:79465 :3748Þ Ce B: Poisson distribution with parameter k ¼ 0 independently a i ¼ð Þ b i ¼ð Þ Ce C: Lognormal distribution with parameter location l ¼ 0 and scale r ¼ 2 independently a i ¼ð0: : :7337 5:30692 : : : : :6237 2: : :3652 0: : :7827Þ b i ¼ð0: : :3904 0:6450 5:6762 : :0086 2: : :8640 9: : :6350 0: :5637Þ Ce D: Gamma distribution with parameter shape c ¼ 2:0 and scale n ¼ :5 independently a i ¼ð6:8924 8:05464 : : : : :93977 :5469 2:2503 5:63549 : :603 :3245 : :85398Þ b i ¼ð2:6298 :429 4:84865 :3704 7: : : :9688 3: :9203 6: : : : :79386Þ Ce E: Exponential distribution with parameter k ¼ 2:0 independently a i ¼ð0:3298 0: : :2935 0:635 : :7082 0: : :345 0:3936 0: :286 0: :06095Þ b i ¼ð0: :8862 0: :42277 : : : : :2445 0: :3308 0:9507 :6380 0:5344 0:302Þ Ce F: Binomial distribution with parameter N ¼ 20, p ¼ 0:3 independently a i ¼ð Þ b i ¼ð Þ The results listed in Tables, 2 and 3 indicate that the A M approximation w more suitable than the normal approximation for the distribution of S n. In support of this, 23

9 Math Sci (205) 9: we observed that the A L approximation w more accurate than the A M approximation in all ces tested. Therefore, we suggest estimating the probability using the A L approximation in ces with large n. Concluding remarks In this paper, we considered both the saddlepoint and moment-bed approximations on the distribution of the sum of i.n.i.d. gamma random variables. Use of the saddlepoint approximation w an accurate method for calculating distribution. From our results, we determined that the precision of the saddlepoint approximation w superior to both the normal and moment-bed approximations. Acknowledgments The author would like to thank the editor and the referee for their valuable comments and suggestions. The author appreciates that this research is supported by the Grant-in-Aid for Young Scientists B of JSPS, KAKENHI Number Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References 2. Daniels, H.E.: Saddlepoint approximations in statistics. Ann. Math. Stat. 25, (954) 3. Daniels, H.E.: Tail probability approximations. Int. Stat. Rev. 55, (987) 4. Eisinga, R., Grotenhuis, M.T., Pelzer, B.: Saddlepoint approximation for the sum of independent non-identically distributed binomial random variables. Stat. Neerl. 67, (203) 5. Gillespie, C.S., Renshaw, E.: An improved saddlepoint approximation. Math. Biosci. 208, (2007) 6. Ha, H.-T., Provost, S.B.: A viable alternative to resorting to statistical tables. Commun. Stat. Simul. Comput. 36, 35 5 (2007) 7. Huzurbazar, S.: Practical saddlepoint approximations. Am. Stat. 53, (999) 8. Jensen, J.L.: Saddlepoint Approximations. Oxford University Press, Oxford (995) 9. Khodabin, M., Ahmadabadi, A.: Some properties of generalized gamma distribution. Math. Sci. 4, 9 28 (200) 0. Kolsa, J.E.: Series Approximation Methods in Statistics. Springer-Verlag, New York (2006). Lugannani, R., Rice, S.O.: Saddlepoint approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab. 2, (980) 2. Mathai, A.M.: Storage capacity of a dam with gamma type inputs. Ann. Inst. Stat. Math. 34, (982) 3. Moschopoulos, P.G.: The distribution of the sum of independent gamma random variables. Ann. Inst. Stat. Math. 37, (985) 4. Murakami, H.: A saddlepoint approximation to the distribution of the sum of independent non-identically uniform random variables. Stat. Neerl. 68, (204) 5. Nadarajah, S., Jiang, X., Chu, J.: A saddlepoint approximation to the distribution of the sum of independent non-identically beta random variables. Stat. Neerl. 69, 02 4 (205). Butler, R.W.: Saddlepoint Approximations with Applications. Cambridge University Press, Cambridge (2007) 23