Density and distribution evaluation for convolution of independent gamma variables. (English) Zbl 1505.62195
Summary: Convolutions of independent gamma variables are encountered in many applications such as insurance, reliability, and network engineering. Accurate and fast evaluations of their density and distribution functions are critical for such applications, but no open source, user-friendly software implementation has been available. We review several numerical evaluations of the density and distribution of convolution of independent gamma variables and compare them with respect to their accuracy and speed. The methods that are based on the Kummer confluent hypergeometric function are computationally most efficient. The benefit of employing the confluent hypergeometric function is further demonstrated by a renewal process application, where the gamma variables in the convolution are Erlang variables. We derive a new computationally efficient formula for the probability mass function of the number of renewals by a given time. An R package coga provides efficient C++ based implementations of the discussed methods and are available in CRAN.
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
Keywords:
confluent hypergeometric function; distribution theory; statistical computing; renewal processReferences:
| [1] | Abramowitz, M.; Stegun, Ia, Handbook of mathematical functions with formulas, graphs, and mathematical tables (1972), New York: Dover, New York · Zbl 0543.33001 |
| [2] | Akkouchi, M., On the convolution of gamma distributions, Soochow J Math, 31, 2, 205-211 (2005) · Zbl 1069.60012 |
| [3] | Barnabani, M., An approximation to the convolution of gamma distributions, Commun Stat Simul Comput, 46, 1, 331-343 (2017) · Zbl 1359.62058 · doi:10.1080/03610918.2014.963612 |
| [4] | Bon, Jl; Pãltãnea, E., Ordering properties of convolutions of exponential random variables, Lifetime Data Anal, 5, 2, 185-192 (1999) · Zbl 0967.60017 · doi:10.1023/A:1009605613222 |
| [5] | Crimaldi, I.; Di Crescenzo, A.; Iuliano, A.; Martinucci, B., A generalized telegraph process with velocity driven by random trials, Adv Appl Probab, 45, 4, 1111-1136 (2013) · Zbl 1291.60182 · doi:10.1239/aap/1386857860 |
| [6] | Di Salvo, F., A characterization of the distribution of a weighted sum of gamma variables through multiple hypergeometric functions, Integral Transforms Spec Funct, 19, 8, 563-575 (2008) · Zbl 1162.33006 · doi:10.1080/10652460802045258 |
| [7] | Furman, E.; Landsman, Z., Risk capital decomposition for a multivariate dependent gamma portfolio, Insurance Math Econ, 37, 3, 635-649 (2005) · Zbl 1129.91025 · doi:10.1016/j.insmatheco.2005.06.006 |
| [8] | Galassi, M.; Davies, J.; Theiler, J.; Gough, B.; Jungman, G., GNU Scientific Library: reference manual (2009), Bristol: Network Theory Ltd, Bristol |
| [9] | Gupta, Ak; Nagar, Dk, Matrix variate distributions (2018), London: Chapman and Hall, London |
| [10] | Hu C, Pozdnyakov V, Yan J (2017) coga: convolution of gamma distributions. https://CRAN.R-project.org/package=coga. R package version 0.2.1 |
| [11] | Iranmanesh, A.; Arashi, M.; Nagar, D.; Tabatabaey, S., On inverted matrix variate gamma distribution, Commun Stat Theory Methods, 42, 1, 28-41 (2013) · Zbl 1298.62085 · doi:10.1080/03610926.2011.577550 |
| [12] | Jain, Gc; Consul, Pc, A generalized negative binomial distribution, SIAM J Appl Math, 21, 4, 501-513 (1971) · Zbl 0234.60010 · doi:10.1137/0121056 |
| [13] | Jasiulewicz, H.; Kordecki, W., Convolutions of Erlang and of Pascal distributions with applications to reliability, Demonstr Math, 36, 1, 231-238 (2003) · Zbl 1017.62105 |
| [14] | Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern actuarial risk theory: using R (2008), Berlin: Springer, Berlin · Zbl 1148.91027 |
| [15] | Kadri, T.; Smaili, K.; Kadry, S.; Kadry, S.; El Hami, A., Markov modeling for reliability analysis using hypoexponential distribution, Numerical methods for reliability and safety assessment: multiscale and multiphysics systems, 599-620 (2015), Cham: Springer, Cham · Zbl 1325.62188 |
| [16] | Khaledi, Be; Kochar, S.; Li, H.; Li, X., A review on convolutions of gamma random variables, Stochastic orders in reliability and risk, 199-217 (2013), Berlin: Springer, Berlin · Zbl 1314.60067 |
| [17] | Mathai, Am, Storage capacity of a dam with gamma type inputs, Ann Inst Stat Math, 34, 1, 591-597 (1982) · Zbl 0505.60099 · doi:10.1007/BF02481056 |
| [18] | Mathai, A., A pathway to matrix-variate gamma and normal densities, Linear Algebra Appl, 396, 317-328 (2005) · Zbl 1084.62044 · doi:10.1016/j.laa.2004.09.022 |
| [19] | Mathai AM, Saxena RK (1978) The H function with applications in statistics and other disciplines. Wiley, New Delhi. http://cds.cern.ch/record/101905 · Zbl 0382.33001 |
| [20] | Moschopoulos, Pg, The distribution of the sum of independent gamma random variables, Ann Inst Stat Math, 37, 1, 541-544 (1985) · Zbl 0587.60015 · doi:10.1007/BF02481123 |
| [21] | Nadarajah, S.; Kotz, S., On the convolution of pareto and gamma distributions, Comput Netw, 51, 12, 3650-3654 (2007) · Zbl 1123.68017 · doi:10.1016/j.comnet.2007.03.003 |
| [22] | Perry, D.; Stadje, W.; Zacks, S., First-exit times for increasing compound processes, Commun Stat Stoch Models, 15, 5, 977-992 (1999) · Zbl 0939.60021 · doi:10.1080/15326349908807571 |
| [23] | Pozdnyakov V, Hu C, Meyer T, Yan J (2017) On discretely observed Brownian motion governed by a continuous time Markov chain. University of Connecticut, Department of Statistics, Tech. rep |
| [24] | R Core Team (2018) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org/ |
| [25] | Sen, A.; Balakrishnan, N., Convolution of geometrics and a reliability problem, Stat Probab Lett, 43, 4, 421-426 (1999) · Zbl 0947.62071 · doi:10.1016/S0167-7152(98)00284-3 |
| [26] | Sim, C., Point processes with correlated gamma interarrival times, Stat Probab Lett, 15, 2, 135-141 (1992) · Zbl 0770.60053 · doi:10.1016/0167-7152(92)90126-P |
| [27] | Srivastava, Hm; Karlsson, Pw, Multiple Gaussian hypergeometric series. Ellis Horwood series in mathematics and its applications (1985), Billingham: E. Horwood, Billingham · Zbl 0552.33001 |
| [28] | Vellaisamy, P.; Upadhye, N., On the sums of compound negative binomial and gamma random variables, J Appl Probab, 46, 1, 272-283 (2009) · Zbl 1161.60303 · doi:10.1017/S0021900200005350 |
| [29] | Zacks, S., Generalized integrated telegraph processes and the distribution of related stopping times, J Appl Probab, 41, 2, 497-507 (2004) · Zbl 1052.60085 · doi:10.1017/S0021900200014455 |
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