Bayesian Inference and Prediction for Normal Distribution Based on Records
Keywords:Bayesian prediction, Best linear unbiased estimators, Maximum likelihood estimators, Record data
Based on record data, the estimation and prediction problems for normal distribution have been investigated by several authors in the frequentist set up. However, these problems have not been discussed in the literature in the Bayesian context. The aim of this paper is to consider a Bayesian analysis in the context of record data from a normal distribution. We obtain Bayes estimators based on squared error and linear-exponential (Linex) loss functions. It is observed that the Bayes estimators can not be obtained in closed forms. We propose using an importance sampling method to obtain Bayes estimators. Further, the importance sampling method is also used to compute Bayesian predictors of future records. Finally, a real data analysis is presented for illustrative purposes and Monte Carlo simulations are performed to compare the performances of the proposed methods. It is shown that Bayes estimators and predictors are superior than frequentist estimators and predictors.
J. AHMADI, M. DOOSTPARAST, A. PARSIAN (2005). Estimation and Prediction in a two-parameter exponential distribution based on k-record values under LINEX loss function. Communications in Statistics-Theory and Methods, 34, pp. 795–805.
M. AHSANULLAH (1995). Record Statistics. Nova Science Publishers, Commack, New York.
B. C. ARNOLD, N. BALAKRISHNAN, H. N. NAGARAJA (1998). Records. John Wiley and Sons, New York.
A. ASGHARZADEH, A. FALLAH (2011). Estimation and prediction for exponentiated family of distributions based on records. Communications in Statistics-Theory and Methods, 40, pp. 68–83.
A. ASGHARZADEH, R. VALIOLLAHI, D. KUNDU (2015). Prediction for future failures in Weibull distribution under hybrid censoring. Journal of Statistical Computation and Simulation, 85, pp. 824–838.
N. BALAKRISHNAN, P. S. CHAN (1998). On the normal record values and associated inference. Statistics and Probability Letters, 39, pp. 73–80.
N. BALAKRISHNAN, A. C. COHEN (1991). Order Statistics and Inference: Estimation Methods. Academic Press, San Diego.
A. P. BASU, N. EBRAHIMI (1991). Bayesian approach to life testing and reliability estimation using a symmetric loss function. Journal of Statistical Planning and Inference, 29, pp. 21–31.
M. CHACKO, M. MARY (2013). Estimation and prediction based on k-record values from normal distribution. Statistica, 73, pp. 505–516.
K.N. CHANDLER (1952). The distribution and frequency of record values. Journal of the Royal Statistical Society: Series B, 14, pp. 220–228.
M. H. CHEN, Q. M. SHAO (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8, pp. 69–92.
R. P. FEYNMAN (1987). Mr. Feynman goes to Washington. Engineering and Science, 51, no. 1, pp. 6–22.
D. KUNDU, H. HOWLADER (2010). Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data. Computational Statistics and Data Analysis, 54, pp. 1547–1558.
D. KUNDU, B. PRADHAN (2009). Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring. Communications in Statistics-Theory and Methods, 38, pp. 2030–2041.
D. KUNDU, M. Z. RAQAB (2012). Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution. Journal of Statistical Planning and Inference, 142, pp. 41–47.
V. B. NEVZOROV (2000). Records: Mathematical Theory. American Mathematical Society, Providence, Rhode Island.
R DEVELOPMENT CORE TEAM (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
M. Z. RAQAB, J. AHMADI, M. DOOSTPARAST (2007). Statistical inference based on record data from Pareto model. Statistics, 41, pp. 105–118.
M. Z. RAQAB, M. T. MADI (2002). Bayesian prediction of the total time on test using doubly censored Rayleigh data. Journal of Statistical Computation and Simulation, 72, pp. 781–789.
C. REN, D. SUN, D. K. DEY (2006). Bayesian and frequentist estimations and prediction for exponential distributions. Journal of Statistical Planning and Inference, 13, pp. 2873–2897.
N. K. SAJEEVKUMAR, M. R. IRSHAD (2014). Estimation of the location parameter of distributions with known coefficient of variation by record values. Statistica, 74, no. 3,pp. 335–349.
A. A. SOLIMAN, F. M. AL-ABOUD (2008). Bayesian inference using record values from Rayleigh model with application. European Journal of Operational Research, 185, pp. 659–672.
H. R. VARRIAN (1975). A Bayesian approach to real estate assessment. In S. E. FEINBERGE, A. ZELLNER (eds.), Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage, North Holland, Amsterdam, pp. 195–208.
S. J. WU, D. H. CHEN, S. T. CHEN (2006). Bayesian inference for Rayleigh distribution under progressive censored sample. Applied Stochastic Models in Business and Industry, 22, pp. 269–279.
A. ZELLNER (1986). Bayesian estimation and prediction using asymmetric loss function. Journal of the American Statistical Association, 81, pp. 446–451.