Skip to main content
Log in

On the distribution of the sum of independent and non-identically extended exponential random variables

  • Original Paper
  • Published:
Japanese Journal of Statistics and Data Science Aims and scope Submit manuscript

Abstract

Considering the sum of the independent and non-identically distributed random variables is a most important topic in many scientific fields. An extension of the exponential distribution based on mixtures of positive distributions is proposed by Gómez et al. (Rev Colomb Estad 37:25–34, 2014). Distribution of the sum of the independent and non-identically distributed random variables is obtained using inverse transformation of the moment generating function. A saddlepoint approximation is used to approximate the derived distribution. Simulations are used to investigate the accuracy of the saddlepoint approximation. Parameters are estimated by the maximum likelihood method. The method is illustrated by the analysis of real data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Almarashi, A. M., Elgarhy, M., Elsehetry, M. M., Golam Kibria, B. M., & Algarni, A. (2019). A new extension of exponential distribution with statistical properties and applications. Journal of Nonlinear Sciences and Applications, 12, 135–145.

    Article  MathSciNet  Google Scholar 

  • Alouini, M.-S., Abdi, A., & Kaveh, M. (2001). Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels. IEEE Transactions on Vehicular Technology, 50, 1471–1480.

    Article  Google Scholar 

  • Amari, S. V., & Misra, R. B. (1997). Closed-form expressions for distribution of sum of exponential random variables. IEEE Transactions on Reliability, 46, 519–522.

    Article  Google Scholar 

  • Bradley, D. M., & Gupta, C. R. (2002). On the distribution of the sum of \(n\) non-identically distributed uniform random variables. Annals of the Institute of Statistical Mathematics, 54, 689–700.

    Article  MathSciNet  Google Scholar 

  • Butler, R. W. (2007). Saddlepoint Approximations with Applications. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Daniels, H. E. (1954). Saddlepoint approximations in statistics. The Annals of Mathematical Statistics, 25, 631–650.

    Article  MathSciNet  Google Scholar 

  • Daniels, H. E. (1987). Tail probability approximations. International Statistical Review, 55, 37–48.

    Article  MathSciNet  Google Scholar 

  • Eisinga, R., Grotenhuis, M. T., & Pelzer, B. (2013). Saddlepoint approximation for the sum of independent non-identically distributed binomial random variables. Statistica Neerlandica, 67, 190–201.

    Article  MathSciNet  Google Scholar 

  • Glaser, R. E. (1983). Statistical analysis of Kevlar 49/epoxy composite stress-rupture data. Livermore: Lawrence Livermore National Laboratory. (Report UCID-19849).

    Google Scholar 

  • Gómez, Y. M., Bolfarine, H., & Gómez, H. W. (2014). A new extension of the exponential distribution. Revista Colombiana de Estadistica, 37, 25–34.

    Article  MathSciNet  Google Scholar 

  • Gupta, R. D., & Kundu, D. (2001). Exponentiated exponential family: An alternative to gamma and Weibull distributions. Biometrical Journal, 43, 117–130.

    Article  MathSciNet  Google Scholar 

  • Huzurbazar, S. (1999). Practical saddlepoint approximations. American Statistician, 53, 225–232.

    Google Scholar 

  • Jensen, J. L. (1995). Saddlepoint approximations. New York: Oxford University Press.

    MATH  Google Scholar 

  • Khuong, H. V., & Kong, H.-Y. (2006). General expression for pdf of a sum of independent exponential random variables. IEEE Communications Letters, 10, 159–161.

    Article  Google Scholar 

  • Kolassa, J. E. (2006). Series approximation methods in statistics. New York: Springer.

    MATH  Google Scholar 

  • Lemonte, A. J., Cordeiro, G. M., & Moreno Arenas, G. (2016). A new useful three-parameter extension of the exponential distribution. Statistics, 50, 312–337.

    MathSciNet  MATH  Google Scholar 

  • Lugannani, R., & Rice, S. O. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables. Advances in Applied Probability, 12, 475–490.

    Article  MathSciNet  Google Scholar 

  • Mathai, A. M. (1982). Storage capacity of a dam with gamma type inputs. Annals of the Institute of Statistical Mathematics, 34, 591–597.

    Article  MathSciNet  Google Scholar 

  • Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics, 37, 541–544.

    Article  MathSciNet  Google Scholar 

  • Murakami, H. (2014). A saddlepoint approximation to the distribution of the sum of independent non-identically uniform random variables. Statistica Neerlandica, 68, 267–275.

    Article  MathSciNet  Google Scholar 

  • Murakami, H. (2015). Approximations to the distribution of sum of independent non-identically gamma random variables. Mathematical Sciences, 9, 205–213.

    Article  MathSciNet  Google Scholar 

  • Nadarajah, S., & Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45, 543–558.

    Article  MathSciNet  Google Scholar 

  • Nadarajah, S., Jiang, X., & Chu, J. (2015). A saddlepoint approximation to the distribution of the sum of independent non-identically beta random variables. Statistica Neerlandica, 69, 102–114.

    Article  MathSciNet  Google Scholar 

  • Olds, E. G. (1952). A note on the convolution of uniform distributions. The Annals of Mathematical Statistics, 23, 282–285.

    Article  MathSciNet  Google Scholar 

  • Sadooghi-Alvandi, S. M., Nematollahi, A. R., & Habibi, R. (2009). On the distribution of the sum of independent uniform random variables. Statistical Papers, 50, 171–175.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Masato Kitani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research of the H. Murakami was supported by JSPS KAKENHI Grant Number 18K11199.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kitani, M., Murakami, H. On the distribution of the sum of independent and non-identically extended exponential random variables. Jpn J Stat Data Sci 3, 23–37 (2020). https://doi.org/10.1007/s42081-019-00046-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42081-019-00046-y

Keywords

Navigation