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Some Properties of the Size-Biased Janardan Distribution

Received: 3 July 2016     Accepted: 19 July 2016     Published: 21 September 2016
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Abstract

Janardan Distribution is one of the important distributions from lifetime models and it has many applications in real life data. A size-biased form of the two parameter Janardan distribution has been introduced in this paper, of which the size-biased Lindley distribution is a special case. Its moments, median, skewness, kurtosis and Fisher index of dispersion are derived and compared with the size-biased Lindley distribution. The shape of the size-biased Janardan distribution is also discussed with graphs. The survival function and hazard rate of the size-biased Janardan distribution have been derived and it is concluded that the hazard rate of the distribution is monotonically increasing. The flexibility in the reliability measures of the size-biased Janardan distribution have been discussed by stochastic ordering. To estimate the parameters of the size-biased Janardan distribution maximum likelihood equations are developed.

Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 5)
DOI 10.11648/j.ajtas.20160505.19
Page(s) 305-310
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Size-Biased Distributions, LD, JD, PJD, SBLD, SBJD, MLE, Stochastic Ordering, IFR

References
[1] Ahmed, A., Reshi, J. A. and Mir, K. A (2013). Structural properties of size-biased Gamma distribution. IOSR Journal of Mathematics, 5 (2), 55-61.
[2] Bashir, S. and Ahmad, M. (2015). Recurrence relations for single and product moments of record values from size-biased Pareto distribution. International Journal of Economic and Business Review, 3 (5), 139-144.
[3] Bashir, S. and Rasul, M. (2015). Some properties of the weighted Lindley distribution. International Journal of Economic and Business Review, 3 (8), 11-17.
[4] Bashir, S. and Rasul, M. (2016). Poisson area-biased Lindley distribution and its applications in biostatistics. Biometrics & Biostatistics International Journal, 3 (1), 1-10.
[5] Ducey, M. J. and Gove, J. H. (2014). Size-biased distributions in the generalized beta family, with applications to forestry. An International Journal of Forest Research, 88 (1), 143-151.
[6] Fisher, R. A. (1934). The effects of methods of ascertainment upon the estimation of frequencies. Ann. Eugenics, 6, 13-25.
[7] Ghitany, M. E. and Al-Mutairi, D. (2008). Size-biased Poisson Lindley distribution and its applications. METRON, LXVI, n. 3, 299-311.
[8] Ghitany, M. E., Atieh, B., and Nadarajah, S. (2008). Lindley distribution and its applications, Mathematics and Computers in Simulation, 78 (4), 493-506.
[9] Lindley, D, V. (1958). Fiducial distributions and Bayes theorem. Journal of Royal Statistical Society, 20, 102-107.
[10] Mir, K. A. and Ahmad, M. (2009). Size-biased distributions and their applications. Pakistan Journal of Statistics, 25 (3), 283-294.
[11] Patil, G. P. and Rao, C. R. (1977). Weighted distributions: a survey of their applications, In P. R. Krishnaiah, (Eds.), Applications of statistics (pp. 383-405). Amsterdam, North-Holland.
[12] Patil, G. P. and Rao, C. R. (1978). Weighted distributions and size-biased sampling with applications to wildlife populations and human families, Biometrics, 34, 179-189.
[13] Rao, C. R. (1965). On discrete distributions arising out of ascertainment, In G. P. Patil (Eds.), Classical and contagious discrete distributions (pp. 320-332). Calcutta, Pergamon Press and Statistical Publishing Society.
[14] Shaked, M., Shanthikumar, J. G. and Valdez-Torres, J. B. (1994). Discrete Probabilistic Orderings in Reliability Theory. StatisticaSinica, 4, 567-579.
[15] Shanker, et al. (2014). The discrete Poisson Janardan distribution with applications. International Journal of Soft Computing and Engineering, 4 (2), 31-33.
[16] Shanker, R. et al (2013). Janardan distribution and its applications to waiting time data. Indian Journal of Applied Research, 3 (8), 500-502.
[17] Shanker, R., Fesshaye, H. and Yemane, A. (2015). On size-biased Poisson-Lindley distribution and its applications to model thunderstorm. American Journal of Mathematics and Statistics, 5 (6), 354-360.
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  • APA Style

    Shakila Bashir, Mujahid Rasul. (2016). Some Properties of the Size-Biased Janardan Distribution. American Journal of Theoretical and Applied Statistics, 5(5), 305-310. https://doi.org/10.11648/j.ajtas.20160505.19

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    ACS Style

    Shakila Bashir; Mujahid Rasul. Some Properties of the Size-Biased Janardan Distribution. Am. J. Theor. Appl. Stat. 2016, 5(5), 305-310. doi: 10.11648/j.ajtas.20160505.19

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    AMA Style

    Shakila Bashir, Mujahid Rasul. Some Properties of the Size-Biased Janardan Distribution. Am J Theor Appl Stat. 2016;5(5):305-310. doi: 10.11648/j.ajtas.20160505.19

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  • @article{10.11648/j.ajtas.20160505.19,
      author = {Shakila Bashir and Mujahid Rasul},
      title = {Some Properties of the Size-Biased Janardan Distribution},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {5},
      number = {5},
      pages = {305-310},
      doi = {10.11648/j.ajtas.20160505.19},
      url = {https://doi.org/10.11648/j.ajtas.20160505.19},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20160505.19},
      abstract = {Janardan Distribution is one of the important distributions from lifetime models and it has many applications in real life data. A size-biased form of the two parameter Janardan distribution has been introduced in this paper, of which the size-biased Lindley distribution is a special case. Its moments, median, skewness, kurtosis and Fisher index of dispersion are derived and compared with the size-biased Lindley distribution. The shape of the size-biased Janardan distribution is also discussed with graphs. The survival function and hazard rate of the size-biased Janardan distribution have been derived and it is concluded that the hazard rate of the distribution is monotonically increasing. The flexibility in the reliability measures of the size-biased Janardan distribution have been discussed by stochastic ordering. To estimate the parameters of the size-biased Janardan distribution maximum likelihood equations are developed.},
     year = {2016}
    }
    

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  • TY  - JOUR
    T1  - Some Properties of the Size-Biased Janardan Distribution
    AU  - Shakila Bashir
    AU  - Mujahid Rasul
    Y1  - 2016/09/21
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    DO  - 10.11648/j.ajtas.20160505.19
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
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    EP  - 310
    PB  - Science Publishing Group
    SN  - 2326-9006
    UR  - https://doi.org/10.11648/j.ajtas.20160505.19
    AB  - Janardan Distribution is one of the important distributions from lifetime models and it has many applications in real life data. A size-biased form of the two parameter Janardan distribution has been introduced in this paper, of which the size-biased Lindley distribution is a special case. Its moments, median, skewness, kurtosis and Fisher index of dispersion are derived and compared with the size-biased Lindley distribution. The shape of the size-biased Janardan distribution is also discussed with graphs. The survival function and hazard rate of the size-biased Janardan distribution have been derived and it is concluded that the hazard rate of the distribution is monotonically increasing. The flexibility in the reliability measures of the size-biased Janardan distribution have been discussed by stochastic ordering. To estimate the parameters of the size-biased Janardan distribution maximum likelihood equations are developed.
    VL  - 5
    IS  - 5
    ER  - 

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Author Information
  • Department of Statistics, Forman Christian College a Chartered University, Lahore, Pakistan

  • Department of Statistics, Forman Christian College a Chartered University, Lahore, Pakistan

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