A Pseudo Estimation of Variance using Prior Information with Unknown Shape Parameter: A Study in Normal Case

Shreya  Bhunia; Babulal  Seal; Proloy  Banerjee

Authors

Shreya Bhunia Department of Mathematics and Statistics, Aliah University, Kolkata, West Bengal, India
Babulal Seal Department of Mathematics and Statistics, Aliah University, Kolkata, West Bengal, India
Proloy Banerjee Department of Mathematics and Statistics, Aliah University, Kolkata, West Bengal, India

Keywords:

Jackknife resampling method, empirical Bayes, risk values

Abstract

In this article, a variant of Bayes estimate for variance parameter from normal distribution is
investigated under weighted squared error loss function. For this purpose, inverse gamma conjugate
prior distribution with an unknown hyper-parameter is considered. Further, empirical Bayes approach
is used to estimate that unknown hyper-parameter from the marginal likelihood equation. We consider some numerical iterative procedure to approximate the hyper-parameter and in this context, the Bayes estimator may be called a pseudo empirical Bayes estimator. To study the performance of the estimator, the integrated risk and bias performance are carried out through an extensive simulation. It is seen that the estimator performs well in accordance with the integrated risk values. To reduce the bias of the estimator, jackknife resampling technique is used further. Finally, the efficiency of the proposed estimator is studied using two real-life datasets and it is found to be satisfactory as they produce lower posterior risk values.

References

Efron B. Empirical Bayes estimates for large-scale prediction problems. J Am Stat Assoc. 2009; 104(487): 1015–1028.

Ferguson TS. Mathematical statistics: A decision theoretic approach. New York and London: Academic Press; 2014.

Banerjee P, Seal B. Partial Bayes Estimation of Two Parameter Gamma Distribution under NonInformative Prior. Stat Optim Inf Comput. 2021; 10(4): 1110-1125.

Bhunia S, Banerjee P. Some Properties and Different Estimation Methods for Inverse A (α) Distribution with an Application to Tongue Cancer Data. Reliab.: Theory Appl. 2022; 17(1 (67)): 251–266.

Ghosh M, Kubokawa T, Kawakubo Y. Benchmarked empirical Bayes methods in multiplicative arealevel models with risk evaluation. Biometrika. 2015; 102(3): 647–659.

Greenland S. Bayesian perspectives for epidemiologic research: III. Bias analysis via missing-data methods. Int J Epidemiol. 2009; 38(6): 1662–1673.

Grabski F, Załe¸ska-Fornal A. Bootstrap methods for the censored data in empirical Bayes estimation of the reliability parameters. Reliab.: Theory Appl. 2010; 5(2 (17)): 115–121.

Hassan AS, Zaky AN. Entropy Bayesian estimation for Lomax distribution based on record. Thail Stat. 2021; 19(1): 95–114.

Jiang W, Zhang CH. General maximum likelihood empirical Bayes estimation of normal means. Ann Stat. 2009; 37(4): 1647–1684.

MacLehose RF, Hamra GB. Applications of Bayesian methods to epidemiologic research. Curr Epidemiol Rep. 2014; 1(3):103-109.

Nichols MD and Padgett WJ. A bootstrap control chart for Weibull percentiles. Qual Reliab Eng Int. 2006; 22(2): 141–151.

Louis TA. Using empirical Bayes methods in biopharmaceutical research. Stat Med. 1991; 10(6): 811–829.

Quenouille MH. Notes on bias in estimation. Biometrika. 1956; 43(3/4): 353–360.

Raghunathan, TE. A quasi-empirical Bayes method for small area estimation. J Am Stat Assoc. 1993; 88(424):1444–1448.

Ristic MM, Kundu D. Marshall-Olkin generalized exponential distribution. Metron. 2015; 73(3): 317–333.

Raweesawat K, Areepong Y, Sukparungsee S, Jampachaisri K. Odds ratio estimation in rare data by empirical Bayes method. Thail Stat. 2017; 15(2): 149–156.

R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2019. http://www.R-project.org/.

Seal B, Hossain SJ. Empirical Bayes estimation of parameters in Markov transition probability matrix with computational methods. J Appl Stat. 2015; 42(3): 508–519.

Tukey J. Bias and confidence in not quite large samples. Ann Math Statist. 1958; 29(2): 614-623.

Tunaru R. Hierarchical Bayesian models for multiple count data. Austrian J Stat. 2002; 31(2&3): 221–229.

Zhang CH. Empirical Bayes and compound estimation of normal means. Stat Sin. 1997; 7(1): 181–193.

Zhang YY, Rong TZ, Li MM. The empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the MLE method. Commun Stat Theory. 2019; 48(9): 2286–2304.