Advertisement

Journal of Statistical Theory and Practice

, Volume 11, Issue 1, pp 26–40 | Cite as

The Maxwell length-biased distribution: Properties and estimation

  • Aamir SaghirEmail author
  • Aneeqa Khadim
  • Zhengyan Lin
Article

Abstract

The concept of length-biased distribution can be employed in the development of proper models for the lifetime data. Length-biased distribution is a special case of the more general form known as the weighted distribution. This article introduces a new class of Maxwell length-biased distribution. The statistical properties of the proposed distribution are determined, including shape, kth moment, reliability function, hazard function, reverse hazard function, and so on. The maximum likelihood estimate (MLE), method of moments (MM) estimate, and Bayesian estimate of the unknown parameter are derived. An illustrative example demonstrates the application of the proposed distribution in real life.

Keywords

Bayes estimate maximum likelihood estimate Maxwell length-biased distribution reliability function sufficient statistic 

AMS Subject Classification

11KXX 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al-khadim, A. K., and A. N. Hussein. 2014. New proposed length biased weighted exponential and Rayleigh distribution with application. Mathematical Theory and Modelling 4:137–52.Google Scholar
  2. Bekker, Α., and J. J. Roux. 2005. Reliability characteristics of the Maxwell distribution: A Bayes estimation study. Communications in Statistics - Theory and Methods 34:2169–78. doi:10.1080/ STA-200066424.MathSciNetCrossRefGoogle Scholar
  3. Chaturvedi, Α., and U. Rani. 1998. Classical and Bayesian reliability estimation of the generalized Maxwell failure distribution. Journal of Statistical Research 32:113–20.MathSciNetGoogle Scholar
  4. Cox, D. R. 1962. Renewal theory. New York, NY: Barnes & Noble.zbMATHGoogle Scholar
  5. Das, Κ. Κ., and Τ. D. Roy. 2011. On some length biased weighted Weibull distribution. Advanced Applied Scientific Research 2:465–75.Google Scholar
  6. Dey, S., and S. S. Maiti. 2010. Bayesian estimation of the parameter of Maxwell distribution under different loss functions. Journal of Statistical Theory and Practice 4:279–87. doi:10.1080/15598608.2010.10411986.MathSciNetCrossRefGoogle Scholar
  7. Fisher, R. A. 1934. The effects of methods of ascertainment upon the estimation of frequencies. Annals of Human Genetics 6:13–25.Google Scholar
  8. Frank, H., and S. C. Althoen. 1994. Statistics concepts and applications. Cambridge, UK: Cambridge University Press.zbMATHGoogle Scholar
  9. Hossain, A. M., and G. Huerta. 2016. Bayesian estimation and prediction for the Maxwell failure distribution based on type II censored data. Open Journal of Statistics 6:49–60. doi:10.4236/ojs.2016.61007.CrossRefGoogle Scholar
  10. Kazmi, Α. Μ., Μ. Aslam, and A. Sajid. 2012. Preference of prior for the class of lifetime distributions under different loss functions. Pakistan Journal Statistics 28:467–84.MathSciNetGoogle Scholar
  11. Kazmi, S. Μ. Α., Μ. Aslam, and A. Sajid. 2011. A note on the maximum likelihood estimators for the mixture of Maxwell distributions using type-i censored scheme. Open Statistics and Probability Journal 3:31–35. doi:10.2174/1876527001103010031.MathSciNetCrossRefGoogle Scholar
  12. Khattree, R. 1989. Characterization of inverse-Gaussian and gamma distributions through their length-biased distributions. IEEE Transactions on Reliability 38:610–11. doi:10.1109/24.46490.CrossRefGoogle Scholar
  13. Lawless, J. F. 2003. Statistical models and methods for life time data (2nd Ed.). New York, NY: John Wiley and Sons.zbMATHGoogle Scholar
  14. Modi, K. 2015. Length-biased weighted Maxwell distribution. Pakistan Journal of Statistics and Operation Research 11:465–72. doi:10.18187/pjsor.vlli4.1008.MathSciNetCrossRefGoogle Scholar
  15. Nanuwong, N., and W. Bodhisuwan. 2014. Length biased beta Pareto distribution and its structural properties with application. Journal of Mathematical Statistics 10:49–57. doi:10.3844/jmssp.2014.49.57.CrossRefGoogle Scholar
  16. Oluyede, B. O., and E. O. George. 2002. On stochastic inequalities and comparisons of reliability measures for weighted distributions. Mathematical Problems in Engineering 8:1–13. doi:10.1080/10241230211380.MathSciNetCrossRefGoogle Scholar
  17. Paul, G. P., and C. R. Rao. 1978. Weighted distributions and size-biased sampling with application to wildlife populations and human families. Biometrics 34:179–89. doi:10.2307/2530008.MathSciNetCrossRefGoogle Scholar
  18. Radha, R. K., and P. Venkatesan. 2013. On the double prior selection for the parameter of Maxwell distribution. International Journal of Scientific & Engineering Research 4:1238–41.Google Scholar
  19. Rao, C. R. 1965. On discrete distributions arising out of methods of ascertainment. In Classical and contagious discrete distributions, ed. G.P. Patil, 320–22. Calcutta: Pergamon Press and Statistical Publishing Society.Google Scholar
  20. Rao, J. Ν. Κ. 1996. On variance estimation with imputed sample data. Journal of American Statistical Association 91:499–506. doi:10.1080/01621459.1996.10476910.CrossRefGoogle Scholar
  21. Saghir, Α., and A. Khadim. 2016. The mathematical properties of length biased Maxwell distribution. Journal of Basic and Applied Research International 16:189–95.Google Scholar
  22. Tomer, S. K., and Panwar, M. S. 2015. Estimation procedures for Maxwell distribution under type I progressive hybrid censoring scheme. Journal of Statistical Computation and Simulation 85:339–56.MathSciNetCrossRefGoogle Scholar
  23. Tyagi, R K., and S. K. Bhattacharya. 1989. A note on the MVU estimation of reliability for the Maxwell failure distribution. Estadística 41:73–79.MathSciNetGoogle Scholar

Copyright information

© Grace Scientific Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsMirpur University of Science and Technology (MUST)MirpurPakistan
  2. 2.Department of Mathematics, Institute of StatisticsZhejiang UniversityHangzhouP.R. China

Personalised recommendations