On the Inverted Gamma Distribution

Salah H. Abid; Saja A. Al-Hassany

doi:doi:10.11648/j.ijssam.20160103.11

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On the Inverted Gamma Distribution

Salah H. Abid, Saja A. Al-Hassany

Published in International Journal of Systems Science and Applied Mathematics (Volume 1, Issue 3)

Received: 29 August 2016 Accepted: 14 September 2016 Published: 28 September 2016

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Abstract

If a random variable follows a particular distribution then the distribution of the reciprocal of that random variable is called inverted distribution. In this paper we studied some issues related with inverted gamma distribution which is the reciprocal of the gamma distribution. We provide forms for the characteristic function, rth raw moment, skewness, kurtosis, Shannon entropy, relative entropy and Rényi entropy function. This paper deals also with the determination of R = P[Y < X] when X and Y are two independent inverted gamma distributions (IGD) with different scale parameters and different shape parameters. Different methods to estimate inverted gamma distribution parameters are studied, Maximum Likelihood estimator, Moments estimator, Percentile estimator, least square estimator and weighted least square estimator. An empirical study is conducted to compare among these methods.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 1, Issue 3)
DOI	10.11648/j.ijssam.20160103.11
Page(s)	16-22
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

Inverted Gamma Distribution, Characteristic Function, Stress-Strength, Shannon Entropy, Relative Entropy, Rényi Entropy, MLE, Percentile Estimator

References

[1]	Abdulah, E. and Elsalloukh, H. (2012) "The Epsilon Skew Inverted Gamma Distribution", JSM Proceeding, Biometrics Section. San Diago California: American Statistical Association.
[2]	Abdulah, E. and Elsalloukh, H. (2014) "Bimodal Class based on the Inverted Symmetrized Gamma Distribution with Applications", J. Stat. Appl. Pro. 3, No. 1, 1-7.
[3]	Ali, M. & Woo, J. & Pal, M. and Wahed, A. (2008) "Some Skew-Symmetric Double Inverted Distributions", International Journal of Statistical Sciences, Vol. 7, pp 1-12.
[4]	Giron, F. and Castello, D. (2001) "A note on the convolution of inverted-gamma distributions with applications to the Behrens-Fisher distribution", Rev. R. Acad. Cien. Serie A. Mat, VOL. 95 (1), pp. 39. 44.
[5]	Kao J (1959). A graphical estimation of mixed Weibull parameters in life testing electron tubes, Technometrics, 1, 389-407.
[6]	Li, T & Peng, L. and Sun, H. (2008( "The Geometric Structure of the Inverse Gamma Distribution", Beitr age zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 49, No. 1, 217-225.
[7]	Llera, A. and Beckmann, C. (2016) "Estimating an Inverse Gamma distribution", Technical report, Radboud University Nijmegen, Donders Institute for Brain Cognition and Behavior.
[8]	Swain J, Venkatraman S and Wilson J (1988). Least squares estimation of distribution function in Johnson's translation system, Journal of Statistical Computation and Simulation, 29, 271-297.
[9]	Witkovsky, V. (2001) "Computing the distribution of a linear combination of inverted gamma variables", Kybernetika 37 (1), p. 79-90.
[10]	Woo, J. (2012) "Inference on Reliability in the Gamma and Inverted Gamma Distributions", PJSOR, Vol. 8, No. 3, pages 635-643, Special Volume In Honour of Distinguished Professor Dr. Mir Masoom Ali.

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APA Style

Salah H. Abid, Saja A. Al-Hassany. (2016). On the Inverted Gamma Distribution. International Journal of Systems Science and Applied Mathematics, 1(3), 16-22. https://doi.org/10.11648/j.ijssam.20160103.11

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ACS Style

Salah H. Abid; Saja A. Al-Hassany. On the Inverted Gamma Distribution. Int. J. Syst. Sci. Appl. Math. 2016, 1(3), 16-22. doi: 10.11648/j.ijssam.20160103.11

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AMA Style

Salah H. Abid, Saja A. Al-Hassany. On the Inverted Gamma Distribution. Int J Syst Sci Appl Math. 2016;1(3):16-22. doi: 10.11648/j.ijssam.20160103.11

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@article{10.11648/j.ijssam.20160103.11,
  author = {Salah H. Abid and Saja A. Al-Hassany},
  title = {On the Inverted Gamma Distribution},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {1},
  number = {3},
  pages = {16-22},
  doi = {10.11648/j.ijssam.20160103.11},
  url = {https://doi.org/10.11648/j.ijssam.20160103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20160103.11},
  abstract = {If a random variable follows a particular distribution then the distribution of the reciprocal of that random variable is called inverted distribution. In this paper we studied some issues related with inverted gamma distribution which is the reciprocal of the gamma distribution. We provide forms for the characteristic function, rth raw moment, skewness, kurtosis, Shannon entropy, relative entropy and Rényi entropy function. This paper deals also with the determination of R = P[Y < X] when X and Y are two independent inverted gamma distributions (IGD) with different scale parameters and different shape parameters. Different methods to estimate inverted gamma distribution parameters are studied, Maximum Likelihood estimator, Moments estimator, Percentile estimator, least square estimator and weighted least square estimator. An empirical study is conducted to compare among these methods.},
 year = {2016}
}

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TY  - JOUR
T1  - On the Inverted Gamma Distribution
AU  - Salah H. Abid
AU  - Saja A. Al-Hassany
Y1  - 2016/09/28
PY  - 2016
N1  - https://doi.org/10.11648/j.ijssam.20160103.11
DO  - 10.11648/j.ijssam.20160103.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 16
EP  - 22
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20160103.11
AB  - If a random variable follows a particular distribution then the distribution of the reciprocal of that random variable is called inverted distribution. In this paper we studied some issues related with inverted gamma distribution which is the reciprocal of the gamma distribution. We provide forms for the characteristic function, rth raw moment, skewness, kurtosis, Shannon entropy, relative entropy and Rényi entropy function. This paper deals also with the determination of R = P[Y < X] when X and Y are two independent inverted gamma distributions (IGD) with different scale parameters and different shape parameters. Different methods to estimate inverted gamma distribution parameters are studied, Maximum Likelihood estimator, Moments estimator, Percentile estimator, least square estimator and weighted least square estimator. An empirical study is conducted to compare among these methods.
VL  - 1
IS  - 3
ER  -

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Author Information

Salah H. Abid

Mathematics Department, Education College, Al-Mustansiriya University, Baghdad, Iraq
Saja A. Al-Hassany

Mathematics Department, Education College, Al-Mustansiriya University, Baghdad, Iraq

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APA Style

Salah H. Abid, Saja A. Al-Hassany. (2016). On the Inverted Gamma Distribution. International Journal of Systems Science and Applied Mathematics, 1(3), 16-22. https://doi.org/10.11648/j.ijssam.20160103.11

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ACS Style

Salah H. Abid; Saja A. Al-Hassany. On the Inverted Gamma Distribution. Int. J. Syst. Sci. Appl. Math. 2016, 1(3), 16-22. doi: 10.11648/j.ijssam.20160103.11

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AMA Style

Salah H. Abid, Saja A. Al-Hassany. On the Inverted Gamma Distribution. Int J Syst Sci Appl Math. 2016;1(3):16-22. doi: 10.11648/j.ijssam.20160103.11

Copy | Download

@article{10.11648/j.ijssam.20160103.11,
  author = {Salah H. Abid and Saja A. Al-Hassany},
  title = {On the Inverted Gamma Distribution},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {1},
  number = {3},
  pages = {16-22},
  doi = {10.11648/j.ijssam.20160103.11},
  url = {https://doi.org/10.11648/j.ijssam.20160103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20160103.11},
  abstract = {If a random variable follows a particular distribution then the distribution of the reciprocal of that random variable is called inverted distribution. In this paper we studied some issues related with inverted gamma distribution which is the reciprocal of the gamma distribution. We provide forms for the characteristic function, rth raw moment, skewness, kurtosis, Shannon entropy, relative entropy and Rényi entropy function. This paper deals also with the determination of R = P[Y < X] when X and Y are two independent inverted gamma distributions (IGD) with different scale parameters and different shape parameters. Different methods to estimate inverted gamma distribution parameters are studied, Maximum Likelihood estimator, Moments estimator, Percentile estimator, least square estimator and weighted least square estimator. An empirical study is conducted to compare among these methods.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - On the Inverted Gamma Distribution
AU  - Salah H. Abid
AU  - Saja A. Al-Hassany
Y1  - 2016/09/28
PY  - 2016
N1  - https://doi.org/10.11648/j.ijssam.20160103.11
DO  - 10.11648/j.ijssam.20160103.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 16
EP  - 22
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20160103.11
AB  - If a random variable follows a particular distribution then the distribution of the reciprocal of that random variable is called inverted distribution. In this paper we studied some issues related with inverted gamma distribution which is the reciprocal of the gamma distribution. We provide forms for the characteristic function, rth raw moment, skewness, kurtosis, Shannon entropy, relative entropy and Rényi entropy function. This paper deals also with the determination of R = P[Y < X] when X and Y are two independent inverted gamma distributions (IGD) with different scale parameters and different shape parameters. Different methods to estimate inverted gamma distribution parameters are studied, Maximum Likelihood estimator, Moments estimator, Percentile estimator, least square estimator and weighted least square estimator. An empirical study is conducted to compare among these methods.
VL  - 1
IS  - 3
ER  -

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