On Bayesian Estimation of Loss and Risk Functions

Randhir Singh

doi:doi:10.11648/j.sjams.20210903.11

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On Bayesian Estimation of Loss and Risk Functions

Randhir Singh

Published in Science Journal of Applied Mathematics and Statistics (Volume 9, Issue 3)

Received: 7 June 2021 Accepted: 21 June 2021 Published: 26 June 2021

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Abstract

Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 9, Issue 3)
DOI	10.11648/j.sjams.20210903.11
Page(s)	73-77
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), 2024. Published by Science Publishing Group

Keywords

Bayes Estimator, Loss Function, Risk Function, H(r, theta) Distribution

References

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[2]	Bhattacharya, S. K.., 1967. Bayesian Approach to Life Testing and Reliability Estimation, JASA 62, pp. 48-62.
[3]	Tyagi, R. K.& Bhattacharya, S. K., 1989. Bayes Esimation of Maxwell’s Velocity Distribution Function, Statistica, anno XLIX, n. 4, pp. 563-567.
[4]	Tyagi, R. K. & Bhattacharya, S. K., 1989. A Note On the MVU Estimation of Reliability for the Maxwell’s Failure Distribution, ESTADISTICA, 41, 137, pp. 73-79.
[5]	Tyagi, R. K. & Bhattacharya, S. K., 1990. Bayesian Survival Analysis Based on the Rayleigh Model, TRABAJOS DE ESTADISTICA, Vol. 5 Num. 1, pp. 81-92.
[6]	Chaturvedi, A. and Rani U., 1998. Classical and Bayesian reliability estimation of the generalized Maxwell failure distribution. Journal of Statistical Research, 32: 113-120.
[7]	Singh, Randhir, 1999. Bayesian Analysis of a Multicomponent System, Proceedings of NSBA-TA, 16-18 Jan. 1999, pp. 252-261, Editor Dr. Rajesh Singh. The conference was organised by the Department of Statistics, Amrawati University, Amravati-444602, Maharashtra, India.
[8]	Singh, Randhir, 2010. Simulation Aided Bayesian Estimation fo Maxwell’s Distribution, Proceedings of National Seminar on Impact of Physics on Biological Sciences (August, 26, 2010), held by the Department of Physics, Ewing Christian College, Prayagraj, India, pp. 203-210; ISBN No.: 978-81-905712-9-6.
[9]	Guobing Fan, 2016. Estimation of the Loss anf Risk Functions of Parameter of Maxwell Distribution. Science Journal of Applied Mathematics and Statistics Vol. 4 No. 4, 2016, pp 129-133. doi: 10.11648/j.sjams.20160404.12.
[10]	Rukhin A. L. 1988. Estimating the loss of estimators of binomial parameter. Biometrica, 75 (1): 153-155.
[11]	Poddar C. K. and Roy M. K. 2003. Bayesian estimation of the parameter of Maxwell distribution under MLINEX loss function, Journal of Statistical Studies, 23: 11-16.
[12]	Bekker, A. and Roux, J. J., 2005. Reliability characteristics of the Maxwell distribution: a Bayes estimation study, Comm. Stat. Theory & Meth., 34 (11): 2169-2178.
[13]	Day S. and Sudhanshu, S. M., 2010. Bayesian estimation of the parameter of Maxwell distribution under different loss functions. Journal of Statistical Theory & Practice, 4 (2): 279-287.
[14]	Krishna H. and Malik M., 2011 Reliability estimation in Maxwell distribution with progressively Type-II censored data. Journal of Statistical Computation and Simulation, 82 (4): 1-19.
[15]	Xu M. P., Ding X. Y. and Yu J., 2013 Bayes inference for the loss and risk functions of Rayleigh distribution parameter estimator. Mathematics in Practice & Theory, 43 (21): 151-156.
[16]	Ajami, Masoud and Jahanshahi, Seyed Mahdi Amir, 2017Journal of modern applied statistical methods: JMASM November 2017. DOI: 10.22237/jmasm/1509495240.
[17]	Tummala, V. M. and Sathe, P. T., 1978. Minimum Expected Loss Estimators of Reliability and Parameters of Certain Life Time Distributions. IEEE Transactions on Reliability, Vol. R-27, No. 4, pp. 283-285.
[18]	Zellner, A. and Park, S. B. 1979. Minimum Expected Loss Estimators (MELO) of Functions of Parameters and Structural Coefficients of Econometric Models. JASA 74, pp. 185-193.
[19]	Teena Goyal, Piyush Kant Rai and Sandeep K. Maurya. 2019. Bayesian Estimation for Exponentiated Weibull distribution under different Loss Functions. International Journal of Pure and Applied Researches. http://ijopar.com2019 Vol. 2 (1): pp. 01-13ISSN: 2455-474X
[20]	Teena Goyal, Piyush Kant Rai and Sandeep K. Maurya. 2020. Bayesian Estimation for Logarithmic Transformed Exponential l distribution under different Loss Functions. Journal of Statistics Applications & Probability. Vol 9 No. 1, pp. 139-148. http://dx.doi.org/10.18576/jsap/090113.

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Randhir Singh. (2021). On Bayesian Estimation of Loss and Risk Functions. Science Journal of Applied Mathematics and Statistics, 9(3), 73-77. https://doi.org/10.11648/j.sjams.20210903.11

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ACS Style

Randhir Singh. On Bayesian Estimation of Loss and Risk Functions. Sci. J. Appl. Math. Stat. 2021, 9(3), 73-77. doi: 10.11648/j.sjams.20210903.11

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AMA Style

Randhir Singh. On Bayesian Estimation of Loss and Risk Functions. Sci J Appl Math Stat. 2021;9(3):73-77. doi: 10.11648/j.sjams.20210903.11

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@article{10.11648/j.sjams.20210903.11,
  author = {Randhir Singh},
  title = {On Bayesian Estimation of Loss and Risk Functions},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {9},
  number = {3},
  pages = {73-77},
  doi = {10.11648/j.sjams.20210903.11},
  url = {https://doi.org/10.11648/j.sjams.20210903.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20210903.11},
  abstract = {Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.},
 year = {2021}
}

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TY  - JOUR
T1  - On Bayesian Estimation of Loss and Risk Functions
AU  - Randhir Singh
Y1  - 2021/06/26
PY  - 2021
N1  - https://doi.org/10.11648/j.sjams.20210903.11
DO  - 10.11648/j.sjams.20210903.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 73
EP  - 77
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20210903.11
AB  - Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.
VL  - 9
IS  - 3
ER  -

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Author Information

Randhir Singh

Department of Statistics, Ewing Christian College, Prayagraj, India

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APA Style

Randhir Singh. (2021). On Bayesian Estimation of Loss and Risk Functions. Science Journal of Applied Mathematics and Statistics, 9(3), 73-77. https://doi.org/10.11648/j.sjams.20210903.11

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ACS Style

Randhir Singh. On Bayesian Estimation of Loss and Risk Functions. Sci. J. Appl. Math. Stat. 2021, 9(3), 73-77. doi: 10.11648/j.sjams.20210903.11

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AMA Style

Randhir Singh. On Bayesian Estimation of Loss and Risk Functions. Sci J Appl Math Stat. 2021;9(3):73-77. doi: 10.11648/j.sjams.20210903.11

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@article{10.11648/j.sjams.20210903.11,
  author = {Randhir Singh},
  title = {On Bayesian Estimation of Loss and Risk Functions},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {9},
  number = {3},
  pages = {73-77},
  doi = {10.11648/j.sjams.20210903.11},
  url = {https://doi.org/10.11648/j.sjams.20210903.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20210903.11},
  abstract = {Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - On Bayesian Estimation of Loss and Risk Functions
AU  - Randhir Singh
Y1  - 2021/06/26
PY  - 2021
N1  - https://doi.org/10.11648/j.sjams.20210903.11
DO  - 10.11648/j.sjams.20210903.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 73
EP  - 77
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20210903.11
AB  - Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.
VL  - 9
IS  - 3
ER  -

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