Skip to main content

A Review on Convolutions of Gamma Random Variables

  • Chapter
  • First Online:
Stochastic Orders in Reliability and Risk

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 208))

Abstract

Due to its wide range of applications, the distribution theory of convolutions of gamma random variables has attracted the attention from many researchers. In this paper, we review some of the latest developments on this problem.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. Amiri, L., Khaledi, B.-E. and Samaniego, F. J.: On skewness and dispersion among convolutions of independent gamma random variables. Probability in the Engineering and Informational Sciences, 25, 55–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bagai, I. and Kochar, S. C.: On tail ordering and comparison of failure rates. Communications in Statistics-Theory and Methods, 15, 1377–1388 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barlow, R. E. and Proschan, F.: Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, Maryland (1981)

    Google Scholar 

  4. Bock, M. E., Diaconis, P., Huffer, F. W. and Perlman, M. D.: Inequalities for linear combinations of Gamma random variables. The Canadian Journal of Statistics, 15, 387–395 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boland, P. J., EI-Newehi, E. and Proschan, F.: Schur properties of convolutions of exponential and geometric random variables. Journal of Multivariate Analysis, 48, 157–167 (1994)

    Google Scholar 

  6. Bon, J. L. and Pǎltǎnea, E.: Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis, 5, 185–192 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diaconis, P. and Perlman, M. D.: Bounds for tail probabilities of linear combinations of independent gamma random variables. The Symposium on dependence in Statistics and Probability, Hidden Valley, Pennsylvania (1987)

    Google Scholar 

  8. Fernández-Ponce, J. M., Kochar, S. C. and Muñoz-Perez, J.: Partial orderings of distributions based on right-spread functions. Journal of Applied Probability, 35, 221–228 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Furman, E.: On a multivariate gamma distribution. Statistics & Probability Letters, 15, 2353–2360 (2008)

    Article  MathSciNet  Google Scholar 

  10. Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M.: Modern Actuarial Risk Theory. Kluwer Academic Publishers (2001)

    Google Scholar 

  11. Khaledi, B.-E.: Stochastic Comparisons of Order Statistics; Spacing and Concomitants of Order Statistics. PhD Thesis, Stat. Math. Unit, Indian Statistical Institute, New Delhi, India (2000)

    Google Scholar 

  12. Khaledi, B.-E. and Kochar, S. C.: Ordering convolutions of gamma random variables. Sankhyā, 66, 466–473 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Kochar, S. C.: On extensions of DMRL and related partial orderings of life distributions. Communications in Statistics—Stochastic Models, 5, 235–245 (1989)

    Google Scholar 

  14. Kochar, S. C., Li, X. and Shaked, M.: The total time on test transform and the excess wealth stochastic order of distributions. Advances in Applied Probability, 34, 826–845 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kochar, S. C. and Ma, C.: Dispersive ordering of convolutions of exponential random variables. Statistics & Probability Letters, 43 321–324 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kochar, S. C. and Xu, M.: On the right spread order of convolutions of heterogeneous exponential random variables. Journal of Multivariate Analysis, 101, 165–176 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kochar, S. C. and Xu, M.: On the skewness of order statistics in the multiple-outlier models. Journal of Applied Probability, 48, 271–284 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kochar, S. C. and Xu, M.: The tail behavior of the convolutions of Gamma random variables. Journal of Statistical Planning and Inference, 141, 418–428 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kochar, S. C. and Xu, M.: Some unified results on comparing linear combinations of independent gamma random variables. Probability in Engineering and Information Sciences, 26, 393–404 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Korwar, R. M.: On stochastic orders for sums of independent random variables. Journal of Multivatiate Analysis, 80, 344–357 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marshall, A. W. and Olkin, I.: Inequalities: theory of majorization and its applications. Academic Press, Inc., New York (1979)

    MATH  Google Scholar 

  22. Marshall, A. W. and Olkin, I.: Life Distributions. Springer, New York (2007)

    MATH  Google Scholar 

  23. Mi, J., Shi, W. and Zhou, Y. Y.: Some properties of convolutions of Pascal and Erlang random variables. Statistics & Probability Letters, 78, 2378–2387 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Serfling, R. J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)

    Book  MATH  Google Scholar 

  25. Shaked, M. and Shanthikumar, J. G.: Stochastic Orders. Springer, New York (2007)

    Book  MATH  Google Scholar 

  26. Shaked, M. and Shanthikumar, J. G.: Stochastic orders and their applications (Second Edition). Springer Series in Statistics. Springer-Verlag, New York (2007)

    Google Scholar 

  27. Yu, Y.: Stochastic ordering of exponential family distributions and their mixtures. Journal of Applied Probability, 46, 244–254 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhao, P.: Some new results on convolutions of heterogeneous gamma random variables. Journal of Multivariate Analysis, 102, 958–976 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhao, P. and Balakrishnan, N.: Mean residual life order of convolutions of heterogenous exponential random variables. Journal of Multivariate Analysis, 100, 1792–1801 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Baha-Eldin Khaledi .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Khaledi, BE., Kochar, S. (2013). A Review on Convolutions of Gamma Random Variables. In: Li, H., Li, X. (eds) Stochastic Orders in Reliability and Risk. Lecture Notes in Statistics(), vol 208. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6892-9_10

Download citation

Publish with us

Policies and ethics