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 |
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Bayes Estimator, Loss Function, Risk Function, H(r, theta) Distribution
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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
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
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
@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} }
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 -