The Poisson Nadarajah-Haghighi Distribution: Different Methods of Estimation

  • Sajid Ali Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan
  • Sanku Dey Department of Statistics, St. Anthony’s College, Shillong 793001, India
  • M H Tahir Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
  • Muhammad Mansoor Department of Statistics, Government Sadiq Egerton College, Bahawalpur, Pakistan
Keywords: Nadarajah-Haghighi distribution, Exponential distribution, hazard rate, maximum likelihood method, Bayesian estimation, Poisson distribution

Abstract

Estimation of parameters of Poisson Nadarajah-Haghighi (PNH) distribution from the frequentist and Bayesian point of view is discussed in this article. To this end, we briefly described ten different frequentist approaches, namely, the maximum likelihood estimators, percentile based estimators, least squares estimators, weighted least squares estimators, maximum product of spacings estimators, minimum spacing absolute distance estimators, minimum spacing absolute-log distance estimators, Cramér-von Mises estimators, Anderson-Darling estimators and right-tail Anderson-Darling estimators. To assess the performance of different estimators, Monte Carlo simulations are done for small and large samples. The performance of the estimators is compared in terms of their bias, root mean squares error, average absolute difference between the true and estimated distribution functions, and the maximum absolute difference between the true and estimated distribution functions of the estimates using simulated data. For the Bayesian inference of the unknown parameters, we use Metropolis–Hastings (MH) algorithm to calculate the Bayes estimates and the corresponding credible intervals. Results from the simulation study suggests that among the considered classical methods of estimation, weighted least squares and the maximum product spacing estimators uniformly produces the least biases of the estimates with least root mean square errors. However, Bayes estimates perform better than all other estimates. Finally, we discuss a practical data set to show the application of the distribution.

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Author Biographies

Sajid Ali, Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan

Sajid Ali is currently Assistant Professor at the Department of Statistics, Quaid-i-Azam University (QAU), Islamabad, Pakistan. He graduated (PhD Statistics) from Bocconi University, Milan, Italy. His research interest is focused on Bayesian inference, construction of new flexible probability distributions, time series analysis, and process monitoring.

Sanku Dey, Department of Statistics, St. Anthony’s College, Shillong 793001, India

Sanku Dey, M.Sc., Ph.D.: An Associate Professor in the Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India. He has to his credit more than 220 research papers in journals of repute. He is a reviewer and associate editors of reputed international journals. He has a good number of contributions in almost all fields of Statistics viz., distribution theory, discretization of continuous distribution, reliability theory, multi-component stress-strength reliability, survival analysis, Bayesian inference, record statistics, statistical quality control, order statistics, lifetime performance index based on classical and Bayesian approach as well as different types of censoring schemes etc.

M H Tahir, Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

M. H. Tahir is currently Professor of Statistics, and Chair Department of Statistics at The University of Bahawalpur (IUB), Bahawalpur, Pakistan. He received BSc, MSc and PhD degree from IUB in 1988, 1990 and 2010, respectively. Dr Tahir has over 27 years of teaching experience to post-graduate classes, and has supervised 65 MPhil and 8 PhD students successfully. He has published more than 90 research papers in national and international journals, including Journal of Statistical Planning and Inference, Communications in Statistics-Theory and Methods, Communications in Statistics-Simulation and Computation, Journal of Statistical Computation and Simulation, Journal of Statistical Distributions and Applications, Journal of Statistical Theory and Applications. He is reviewer of more than 55 national and international statistical journals. Dr. Tahir’s research interests include distribution theory, generalized classes of distributions, survival and lifetime data analysis, methods of estimation, and construction of experimental designs.

Muhammad Mansoor, Department of Statistics, Government Sadiq Egerton College, Bahawalpur, Pakistan

Muhammad Mansoor is Assistant Professor of Statistics at the Department of Statistics, Government Sadiq Egerton Graduate College, Bahawalpur, Pakistan. His current research focuses on generalizing statistical distributions arising from the hazard function. Other research areas include statistical inference of probability models, computational statistics, and regression analysis.

References

Aarset (1987) Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, R-36(1):106–108.

Ali et al. (2020a) Ali, S., Dey, S., Tahir, M. H., and Mansoor, M. (2020a). A comparison of different methods of estimation for the flexible Weibull distribution. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69:794–814.

Ali et al. (2020b) Ali, S., Dey, S., Tahir, M. H., and Mansoor, M. (2020b). Two-parameter logistic-exponential distribution: Some new properties and estimation methods. American Journal of Mathematical and Management Sciences, 39(3):270–298.

Ali et al. (2020c) Ali, S., Dey, S., Tahir, M. H., and Mansoor, M. (2020c). Two-Parameter Logistic-Exponential Distribution: Some New Properties and Estimation Methods. American Journal of Mathematical and Management Sciences, 39(3):270–298.

Alizadeh et al. (2020) Alizadeh, M., Afify, A. Z., Eliwa, M. S., and Ali, S. (2020). The odd log-logistic Lindley-G family of distributions: properties, Bayesian and non-Bayesian estimation with applications. Computational Statistics, 35:281–308.

Anderson and Darling (1952) Anderson, T. W. and Darling, D. A. (1952), Asymptotic Theory of Certain “Goodness of Fit” Criteria Based on Stochastic Processes, The Annals of Mathematical Statistics, 23(2):193–212. https://doi.org/10.1214/aoms/1177729437

Chen and Balakrishnan (1995) Chen, G. and Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27(2):154–161.

Cheng and Amin (1979) Cheng, R. C. H. and Amin, N. A. K. (1979). Maximum product of spacings estimation with application to the lognormal distribution, math.

Cheng and Amin (1983) Cheng, R. C. H. and Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society. Series B (Methodological), 45(3):394–403.

Devroye (1986) Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer-Verlag.

Dey et al. (2017a) Dey, S., Al-Zahrani, B., and Basloom, S. (2017a). Dagum distribution: Properties and different methods of estimation. International Journal of Statistics and Probability, 6(2):74–92.

Dey et al. (2015) Dey, S., Ali, S., and Park, C. (2015). Weighted exponential distribution: properties and different methods of estimation. Journal of Statistical Computation and Simulation, 85(18):3641–3661.

Dey et al. (2016) Dey, S., Dey, T., Ali, S., and Mulekar, M. S. (2016). Two-parameter Maxwell distribution: Properties and different methods of estimation. Journal of Statistical Theory and Practice, 10(2):291–310.

Dey et al. (2014) Dey, S., Dey, T., and Kundu, D. (2014). Two-parameter Rayleigh distribution: Different methods of estimation. American Journal of Mathematical and Management Sciences, 33(1):55–74.

Dey et al. (2017b) Dey, S., Kumar, D., Ramos, P. L., and Louzada, F. (2017b). Exponentiated Chen distribution: Properties and estimation. Communications in Statistics – Simulation and Computation, 46(10):8118–8139.

Dey et al. (2017c) Dey, S., Raheem, E., and Mukherjee, S. (2017c). Statistical Properties and Different Methods of Estimation of Transmuted Rayleigh Distribution. Revista Colombiana de EstadÃstica, 40:165–203.

Dey et al. (2017d) Dey, S., Raheem, E., Mukherjee, S., and Ng, H. K. T. (2017d). Two parameter exponentiated Gumbel distribution: properties and estimation with flood data example. Journal of Statistics and Management Systems, 20(2):197–233.

Eliwa et al. (2020) Eliwa, M. S., El-Morshedy, M., and Ali, S. (2020). Exponentiated odd Chen-G family of distributions: statistical properties, Bayesian and non-Bayesian estimation with applications. Journal of Applied Statistics, 0(0):1–27.

Ghitany et al. (2005) Ghitany, M. E., Al-Hussaini, E. K., and Al-Jarallah, R. A. (2005). Marshall–Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics, 32(10):1025–1034.

Gupta and Kundu (1999) Gupta, R. D. and Kundu, D. (1999). Theory & methods: Generalized exponential distributions. Australian & New Zealand Journal of Statistics, 41(2):173–188.

Kao (1958) Kao, J. H. K. (1958). Computer methods for estimating Weibull parameters in reliability studies. IRE Transactions on Reliability and Quality Control, PGRQC-13:15–22.

Kao (1959) Kao, J. H. K. (1959). A graphical estimation of mixed Weibull parameters in life-testing of electron tubes. Technometrics, 1(4):389–407.

Kenney and Keeping (1962) Kenney, J. and Keeping, E. (1962). Mathematics of statistics. Number v. 2 in Mathematics of Statistics. Princeton: Van Nostrand.

Kundu and Raqab (2005) Kundu, D. and Raqab, M. Z. (2005). Generalized Rayleigh distribution: different methods of estimations. Computational Statistics & Data Analysis, 49(1):187–200.

Lee and Wang (2013) Lee, E. T. and Wang, J. W. (2013). Statistical Methods for Survival Data Analysis. Wiley Publishing, 4th edition.

Lehmann and Casella (2003) Lehmann, E. and Casella, G. (2003). Theory of Point Estimation. Springer New York.

Lemonte (2013) Lemonte, A. J. (2013). A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Computational Statistics & Data Analysis, 62:149–170.

MacDonald (1971) MacDonald, P. D. M. (1971). Comment on “an estimation procedure for mixtures of distributions” by Choi and Bulgren. Journal of the Royal Statistical Society. Series B (Methodological), 33(2):326–329.

Mansoor et al. (2020a) Mansoor, M., Tahir, M. H., Alzaatreh, A., and Cordeiro, G. M. (2020a). The Poisson Nadarajah–Haghighi distribution: Properties and applications to lifetime data. International Journal of Reliability, Quality and Safety Engineering, 27(01):2050005.

Mansoor et al. (2020b) Mansoor, M., Tahir, M. H., Cordeiro, G. M., Ali, S., and Alzaatreh, A. (2020b). The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data. Mathematica Slovaca, 70(4):917–934.

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

Moors (1988) Moors, J. J. A. (1988). A quantile alternative for kurtosis. Journal of the Royal Statistical Society. Series D (The Statistician), 37(1):25–32.

Mudholkar and Srivastava (1993) Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42(2):299–302.

Nadarajah and Haghighi (2011) Nadarajah, S. and Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6):543–558.

Nadarajaha and Kotz (2006) Nadarajaha, S. and Kotz, S. (2006). The beta exponential distribution. Reliability Engineering & System Safety, 91(6):689–697.

Ranneby (1984) Ranneby, B. (1984). The maximum spacing method. an estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics, 11(2):93–112.

Shafqat et al. (2020) Shafqat, M., Ali, S., Shah, I., and Dey, S. (2020). Univariate discrete Nadarajah and Haghighi distribution: Properties and different methods of estimation. Statistica, 80(3):301–330.

Swain et al. (1988) Swain, J. J., Venkatraman, S., and Wilson, J. R. (1988). Least-squares estimation of distribution functions in Johnson’s translation system. Journal of Statistical Computation and Simulation, 29(4):271–297.

Tahir et al. (2018) Tahir, M. H., Cordeiro, G. M., Ali, S., Dey, S., and Manzoor, A. (2018). The inverted Nadarajah–Haghighi distribution: estimation methods and applications. Journal of Statistical Computation and Simulation, 88(14):2775–2798.

Teimouri et al. (2013) Teimouri, M., Hoseini, S. M., and Nadarajah, S. (2013). Comparison of estimation methods for the Weibull distribution. Statistics, 47(1):93–109.

Torabi (2008) Torabi, H. (2008). A general method for estimating and hypotheses testing using spacings. Journal of Statistical Theory and Practice, 8(2):163–168.

Published
2021-08-30
Section
Articles