Based on a record sample from the Rayleigh model, we consider the problem of estimating the scale and location parameters of the model and predicting the future unobserved record data. Maximum likelihood and Bayesian approaches under different loss functions are used to estimate the model’s parameters. The Gibbs sampler and Metropolis-Hastings methods are used within the Bayesian procedures to draw the Markov Chain Monte Carlo (MCMC) samples, used in turn to compute the Bayes estimator and the point predictors of the future record data. Monte Carlo simulations are performed to study the behaviour and to compare methods obtained in this way. Two examples of real data have been analyzed to illustrate the procedures developed here.

Bayesian estimation and prediction, Rayleigh distribution, record values, Markov Chain Monte Carlo samples

Abu Awwad, R. R., Bdair, O. M. and Abufoudeh, G. K., (2018). One and Two-Sample Prediction for the Progressively Censored Rayleigh Residual Data. Journal of Statistical Theory and Practice, 12(4), pp. 669–687.

Ahmadi, J. and Doostparast, M., (2006). Bayesian estimation and prediction for some life distributions based on record values. Statistical Papers, 47(3), pp. 373–392.

Arnold, B. C, Balakrishnan, N. and Nagaraja, H. N., (1998). Records. John Weily, New York.

Ahsanullah, M., (1980). Linear prediction of record values for the two parameter exponential distributrion. Annals of the Institute of Statistical Mathematics, 32, pp. 363–368.

Ahsanullah, M., (2004). Record Values-Theory and Applications. University Press of America, New York.

Al-Hussaini, E. K. and Ahmad, A. A., (2003). On Bayesian interval prediction of future records. Test, 12, pp. 79–99.

Bdair, O.M. and Raqab, M. Z., (2009). On the mean residual waiting time of records. Statistics & Decisions International mathematical journal for stochastic methods and models, 27(3), pp. 249-–260, DOI: https://doi.org/10.1524/stnd.2009.1050.

Bdair, O. M. and Raqab, M. Z., (2016). One-sequence and two-sequence prediction for future Weibull records. Journal of Statistical Theory and Applications, 15(4), pp. 345– 366.

Berger, J. O. and Sun., D., (1993). Bayesian analysis for the poly-Weibull distribution. Journal of the American Statistical Association, 88(424), pp. 1412-–1418. DOI:10.1080/01621459.1993.10476426.

David Hinkley, (1977). On quick choice of power transformation. Journal of the Royal Statistical Society Series C, 26(1), pp. 67–69.

Gulati, S. and Padgett, W. J., (2003). Parametric and nonparametric inference from recordbreaking data. Springer-Verlag, New York.

Johnson, N. L, Kotz, S. and Balakrishnan, N., (1994). Continuous univariate distribution. 2-nd edition. Wiley and Sons, New York.

Jung In Seo. and Yongku Kim, (2018). Objective Bayesian inference based on upper record values from Rayleigh distribution . Communications for Statistical Applications and Methods, 25(4), pp. 411—430.

Kundu, D., (2008). Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring. Technometrics, 50, pp. 144–154.

Lawless, J.F., (1982). Statistical models and methods for lifetime data. 2nd Edition, Wiley, New York.

Madi, M. T. and Raqab, M. Z., (2004). Bayesian prediction of temperature records using the Pareto model. Environmetrics, 15, pp. 701–710.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E., (1953). Equations of State Calculations by Fast Computing Machines. Journal Chemical Physics, 21, pp. 1087–1091.

Nagaraja, H. N., (1984). Asymptotic linear prediction of extreme order statistics. Annals of the Institute of Statistical Mathematics, 36, pp. 289–299.

Nevzorov, V. B., (2000). Records: Mathematical Theory (English Translation). American Mathematical Society, Providence, Rhode Island.

Raqab, M. Z., Ahmadi, J. and Doostparast, M. (2007). Statistical inference based on record data from Pareto model. Statistics, 42(2), pp. 105–118.

Raqab, M. Z., Bdair, O. M. and Al-Aboud, F. M., (2018). Inference for the two-parameter bathtub-shaped distribution based on record data. Metrika, 81(3), pp. 229—253.

Rayleigh, L., (1880). On the resultant of a large number of vibrations of same pitch and of arbitrary phase. Philosophical Magazine, 10, pp. 73-–78.

Soliman, A. A. and Al-Aboud, F. M., (2008). Bayesian inference using record values from Rayleigh model with application. European Journal of Operational Research, 185(2), pp. 659–672.

Varian, H. R., (1975). A Bayesian approach to real estate assessment. Studies in Bayesian Econometrics and Statistics in Honor of L.J. Savage. North Holland, Amsterdam, pp. 195–208.

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