Analysis for the xgamma distribution based on record values and inter-record times with application to prediction of rainfall and COVID-19 records

Zahra Khoshkhoo Amiri; S. M. T. K. MirMostafaee

doi:https://doi.org/10.59170/stattrans-2023-065

Analysis for the xgamma distribution based on record values and inter-record times with application to prediction of rainfall and COVID-19 records

Zahra Khoshkhoo Amiri Department of Statistics, University of Mazandaran, Babolsar, Iran ORCID:https://orcid.org/0000-0001-8838-5870 , S. M. T. K. MirMostafaee 2Department of Statistics, University of Mazandaran, Babolsar, Iran ORCID:https://orcid.org/0000-0003-2796-4427 Statistics in Transition new series, vol. 24, 2023, 5, pages: 89-108 Published online: 7 December 2023 https://doi.org/10.59170/stattrans-2023-065

214 Views 29 Downloads

ARTICLE

(English) PDF

ABSTRACT

Recently, Sen et al. (2016) introduced a new lifetime distribution, called “xgamma distribution", which can be used as an alternative to other lifetime distributions, like the exponential one. In this paper, we study the problem of classical and Bayesian estimation of the unknown parameter of the xgamma distribution based on record values and inter-record times. The problem of Bayesian prediction of future record values based on record values and interrecord times is also discussed. A small simulation study has been performed to compare the performance of the proposed estimators and the approximate Bayes predictors. Two real data sets related to rainfall and COVID-19 records have been analysed. We considered four one-parameter lifetime distributions as the base models for each data set and compared the goodness-of-fit results. Then, the numerical results of estimation of the parameter and prediction of future records based on the xgamma and exponential records and inter-record times were presented. We observed that the record values and inter-record times from the xgamma distribution could predict future records in a relatively satisfactory way.

KEYWORDS

COVID-19 records, lower record values, Bayes predictive distribution, rainfall records, xgamma distribution.

REFERENCES

Ahmadi, J., MirMostafaee, S. M. T. K., (2009). Prediction intervals for future records and order statistics coming from two parameter exponential distribution. Statistics & Probability Letters, 79(7), pp. 977–983.

AL-Hussaini, E.K., Al-Awadhi, F., (2010). Bayes two-sample prediction of generalized order statistics with fixed and random sample size. Journal of Statistical Computation and Simulation, 80(1), pp. 13–28.

Albert, J., (2009). Bayesian Computation with R, 2nd Ed., LLC: Springer.

Amini, M., MirMostafaee, S. M. T. K., (2016). Interval prediction of order statistics based on records by employing inter-record times: A study under two parameter exponential distribution. Metodološki Zvezki, 13(1), pp. 1–15.

Arnold, B. C., Balakrishnan, N., Nagaraja, H. N., (1998). Records, New York: John Wiley & Sons.

Bastan, F., MirMostafaee, S. M. T. K., (2022). Estimation and prediction for the Poissonexponential distribution based on records and inter-record times: A comparative study. Journal of Statistical Sciences, 15(2), 381–405.

Chen, M.-H., Shao, Q.-M., (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), pp. 69–92.

El-Monsef, M. M. E. A., Sweilam, N.H., Sabry, M. A., (2021). The exponentiated power Lomax distribution and its applications. Quality and Reliability Engineering International, 37(3), pp. 1035–1058.

Fallah, A., Asgharzadeh, A., MirMostafaee, S. M. T. K., (2018). On the Lindley record values and associated inference. Journal of Statistical Theory and Applications, 17(4), pp. 686–702.

Geweke, J., (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bayesian Statistics 4, Eds. J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, Clarendon Press, Oxford, UK, pp. 169–193.

Ghitany, M. E., Atieh, B., Nadarajah, S., (2008). Lindley distribution and its application. Mathematics and Computers in Simulation, 78(4), pp. 493–506.

Hasselman, B., (2018). nleqslv: Solve systems of nonlinear equations, R package version 3.3.2, https://CRAN.R-project.org/package=nleqslv.

Hastings, W. K., (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), pp. 97–109.

Heidelberger, P., Welch, P. D., (1981a). A spectral method for confidence interval generation and run length control in simulations. Communications of the ACM, 24(4), pp. 233–245. Heidelberger, P., Welch, P. D., (1981b). Adaptive spectral methods for simulation output analysis. IBM Journal of Research and Development, 25(6), pp. 860–876.

Heidelberger, P., Welch, P. D., (1983). Simulation run length control in the presence of an initial transient. Operations Research, 31(6), pp. 1109–1144.

Kizilaslan, F., Nadar, M., (2015). Estimation with the generalized exponential distribution based on record values and inter-record times. Journal of Statistical Computation and Simulation, 85(5), pp. 978–999.

Kumar, D., Dey, S., Ormoz, E., MirMostafaee, S. M. T. K., (2020). Inference for the unit-Gompertz model based on record values and inter-record times with an application. Rendiconti del Circolo Matematico di Palermo Series 2, 69(3), pp. 1295–1319.

Lehmann, E. L., Casella, G., (1998). Theory of Point Estimation, 2nd Ed., New York: Springer.

Lindley, D. V., (1958). Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Series B (Methodological), 20(1), pp. 102–107.

Marinho, P. R. D., Bourguignon, M., Dias, C. R. B., (2013). AdequacyModel: Adequacy of probabilistic models and generation of pseudo-random numbers, R package version 1.0.8., https://CRAN.R-project.org/package=AdequacyModel.

Mersmann, O., Trautmann, H., Steuer, D., Bornkamp, B., (2018). truncnorm: Truncated normal distribution, R package version 1.0-8, https://CRAN.R-project.org/package=truncnorm.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., Teller, E., (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), pp. 1087–1092.

MirMostafaee, S. M. T. K., Asgharzadeh, A., Fallah, A., (2016). Record values from NH distribution and associated inference. Metron, 74(1), pp. 37–59.

Nadar, M., Kizilaslan, F., (2015). Estimation and prediction of the Burr type XII distribution based on record values and inter-record times. Journal of Statistical Computation and Simulation, 85(16), pp. 3297–3321.

Pak, A., Dey, S., (2019). Statistical inference for the power Lindley model based on record values and inter-record times. Journal of Computational and Applied Mathematics, 347, pp. 156–172.

Plummer, M., Best, N., Cowles, K., Vines, K., (2006). CODA: Convergence diagnosis and output analysis for MCMC. R News, 6(1), pp. 7–11.

Plummer, M., Best, N., Cowles, K., Vines, K., Sarkar, D., Bates, D., Almond, R., Magnusson, A., (2018). coda: Output analysis and diagnostics for MCMC, R package version 0.19-2, https://CRAN.R-project.org/package=coda.

R Core Team, (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Raftery, A. E., Lewis, S. M., (1992). Comment: One long run with diagnostics: Implementation

strategies for Markov chain Monte Carlo. Statistical Science, 7(4), pp. 493–497.

Raftery, A. E., Lewis, S. M., (1996). Implementing MCMC. In Markov Chain Monte Carlo in Practice, Eds. W.R. Gilks, S. Richardson and D.J. Spiegelhalter, Chapman and Hall/CRC, Boca Raton, pp. 115–130.

Samaniego, F. J., Whitaker, L. R., (1986). On estimating population characteristics from record-breaking observations. i. parametric results. Naval Research Logistics Quarterly, 33(3), pp. 531–543.

Schruben, L. W., (1982). Detecting initialization bias in simulation output. Operations Research, 30(3), pp. 569–590.

Schruben, L., Singh, H., Tierney, L., (1980). A test of initialization bias hypotheses in simulation output. Technical Report 471, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 14853.

Schruben, L., Singh, H., Tierney, L., (1983). Optimal tests for initialization bias in simulation output. Operations Research, 31(6), pp. 1167–1178.

Sen, S., Chandra, N., Maiti, S. S., (2018). Survival estimation in xgamma distribution under progressively type-II right censored scheme. Model Assisted Statistics and Applications, 13(2), pp. 107–121.

Sen, S., Maiti, S. S., Chandra, N., (2016). The xgamma distribution: Statistical properties and application. Journal of Modern Applied Statistical Methods, 15(1), pp. 774–788.

Shanker, R., (2015). Shanker distribution and its applications. International Journal of Statistics and Applications, 5(6), pp. 338–348.

Tierney, L., Kadane, J. B., (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81(393), pp. 82– 86.

Varian, H. R., (1975). Bayesian approach to real estate assessment. In Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage, Eds. S.E. Fienberg and A. Zellner, North-Holland Pub. Co., Amsterdam, pp. 195–208.

Yadav, A. S., Saha, M., Singh, S. K., Singh, U., (2019). Bayesian estimation of the parameter and the reliability characteristics of xgamma distribution using type-II hybrid censored data. Life Cycle Reliability and Safety Engineering, 8(1), pp. 1–10.

Zellner, A., (1986). Bayesian estimation and prediction using asymmetric loss functions. Journal of the American Statistical Association, 81(394), pp. 446–451.