Some Properties of the Size-Biased Janardan Distribution

Shakila Bashir; Mujahid Rasul

doi:doi:10.11648/j.ajtas.20160505.19

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Some Properties of the Size-Biased Janardan Distribution

Shakila Bashir, Mujahid Rasul

Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 5)

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
PY  - 2016
N1  - https://doi.org/10.11648/j.ajtas.20160505.19
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
SP  - 305
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

Shakila Bashir

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

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

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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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Some Properties of the Size-Biased Janardan Distribution
AU  - Shakila Bashir
AU  - Mujahid Rasul
Y1  - 2016/09/21
PY  - 2016
N1  - https://doi.org/10.11648/j.ajtas.20160505.19
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
SP  - 305
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  -

Copy | Download