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Estimation of the location and the scale parameters of Burr Type XII distribution

Year 2019, Volume: 68 Issue: 1, 1030 - 1044, 01.02.2019

Fatma Gül Akgül , Şükrü Acıtaş Birdal Şenoğlu

https://doi.org/10.31801/cfsuasmas.501455

Abstract

The aim of this paper is to estimate the location and the scale parameters of Burr Type XII distribution. For this purpose, different estimation methods, namely, maximum likelihood (ML), modified maximum likelihood (MML), least squares (LS) and method of moments (MM) are used. The performances of these estimation methods are compared via Monte-Carlo simulation study under different sample sizes and parameter settings. At the end of the study, the wind speed data set and the annual flow data sets are analyzed for illustration of the modeling performance of Burr Type XII distribution. 

Keywords

Burr XII distribution maximum likelihood modified maximum likelihood least squares method of moments efficiency

References

  • Abbasi, B., Hosseinifard, S. Z. and Coit, D. W., A neural network applied to estimate Burr XII distribution parameters, Reliability Engineering and System Safety 95, (2010), 647--654.
  • Abd-Elfattah, A. M., Hassan, A. S. and Nassr, S. G., Estimation in step-stress partially accelerated life tests for the Burr type XII distribution using type I censoring, Statistical Methodology 5, (2008), 502--514.
  • Acitas, S., Kasap, P., Senoglu, B. and Arslan, O., One-step M-estimators: Jones and Faddy's skewed t-distribution, Journal of Applied Statistics 40(7), (2013), 1545--1560.
  • Abuzaid, A. H., The estimation of the Burr-XII parameters with middle-censored data, SpringerPlus 4(101), (2015), 1--10.
  • Afify, A. Z., Cordeiro, G. M., Ortega, E. M. M., Yousof H. M. and Butt, N. S., The four-parameter Burr XII distribution: Properties, regression model and applications, Communications in Statistics-Theory and Methods 47(11), (2018), 2605--2624.
  • Akgul, F. G., and Senoglu, B., Burr XII daglmnn konum ve olcek parametrelerinin tahmini: Monte Carlo Simulasyon Calısması, ISTKON8, Antalya, Turkey.
  • Akgul, F. G., Senoglu, B. and Arslan, T., An alternative distribution to Weibull for modeling the wind speed data: Inverse Weibull distribution, Energy Conversion and Management 114, (2016), 234-240.
  • Arslan, T., Acitas, S., and Senoglu, B., Generalized Lindley and Power Lindley distributions for modeling the wind speed data, Energy Conversion and Management 152, (2017), 300--311.
  • Aydin, D., Akgul, F. G. and Senoglu, B., Robust estimation of the location and the scale parameters of shifted Gompertz distribution, Electronic Journal of Applied Statistical Analysis 11(1), (2018), 92--107.
  • Bowman, K. O. and Shenton, L. R., Weibull distributions when the shape parameter is defined, Computational Statistics and Data Analysis 36, (2001), 299--310.
  • Brano, V. L., Orioli, A., Ciulla, G. and Culotta, S., Quality of wind speed fitting distributions for the urban area of Palermo, Italy, Renewable Energy 36, (2011), 1026--1039.
  • Burr, I. W., Cumulative frequency functions, Annals of Mathematical Statistics, 13, (1942), 215--232.
  • Chou, C. -Y., Chen, C. -H. and Liu, H. -R., Acceptance Control Charts for Non-normal Data, Journal of Applied Statistics 32(1), (2005), 25--36.
  • Dogru, F. Z. and Arslan, O., Optimal B-robust estimators for the parameters of the Burr XII distribution, Journal of Statistical Computation and Simulation 86(6), (2016), 1133--1149.
  • Gomes, A. E., da-Silva, C. Q. and Cordeiro, G. M., Two extended Burr models: Theory and practice, Commun. Stat. Theory - Methods 44, (2015), 1706--1734.
  • Hossain, A. M. and Nath, S. K., Estimation of parameters in the presence of outliers for a burr XII distribution, Communications in Statistics-Theory and Methods 26(3), (1997), 637--652.
  • slam, M. Q. and Tiku, M. L., Multiple linear regression model under nonnormality, Communications in Statistics-Theory and Methods 33(10), (2004), 2443--2467.
  • Kantar, Y. M. and Senoglu, B., A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter, Computers and Geosciences 34(12), (2008), 1900--1909.
  • Klugman, S. A., Panjer, H. H. and Willmot, G. E., Loss Models: From Data to Decisions, Wiley, New York, 1998.
  • Mead, M. E. and Afify, A. Z., On five parameter Burr XII distribution: Properties and applications, South Afr. Stat. J. 51, (2017), 67-80.
  • Nadar, M. and Papadopoulos, A. S., Bayesian analysis for the Burr Type XII distribution based on record values, Statistica 71(4), (2011), 421--435.
  • Panahi, H. and Sayyareh, A. A., Parameter estimation and prediction of order statistics for the Burr Type XII distribution with Type II censoring, Journal of Applied Statistics 41(1), (2014), 215--232.
  • Papalexiou, S. M. and Koutsoyiannis, D., Entropy based derivation of probability distributions: A case study to daily rainfall, Advances in Water Resources 45, (2012) 51--57.
  • Puthenpura, S. and Sinha, N. K., Modified maximum likelihood method for the robust estimation of system parameters from very noisy data, Automatica 22, (1986), 231--235.
  • Rao, G. S., Aslam, M. and Kundu, D., Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength, Communications in Statistics â€" Theory and Methods 44, (2015), 4953--4961.
  • Rodriguez, N., A guide to the Burr type XII distributions, Biometrika 64, (1977), 129--134.
  • Shao, Q., Wong, H., Xia, J. and Ip, W. C., Models for extremes using the extended three- parameter Burr XII system with application to flood frequency analysis, Hydrological Sciences 49, (2004), 685â€"-702.
  • Singh, S. K. and Maddala, G. S., A function for the size distribution of income, Econometrica 44, (1976), 963--970.
  • Soliman, A. A., Estimation of parameters of life from progressively censored data using Burr-XII model, IEEE Transactions on Reliability, 54, (2005), 34--42.
  • Soliman, A. A., Abd Ellah, A. H., Abou-Elheggag, N. A. and Modhesh, A. A., Estimation from Burr type XII distribution using progressive first-failure censored data, Journal of Statistical Computation and Simulation 83(12), (2013), 2270--2290.
  • Şenoğlu, B., Estimating parameters in one-way analysis of covariance model with short-tailed symmetric error distributions, Journal of Computational and Applied Mathematics 201, (2007), 275--283.
  • Swain, J., Venkatraman, S. and Wilson, J., Least squares estimation of distribution function in Johnson’s translation system, Journal of Statistical Computation and Simulation 29, (1988), 271â€"-297.
  • Tadikamalla, P. R., A look at the Burr and related distributions, International Statistical Review 48, (1989), 337--344.
  • Tiku, M. L., Estimating the mean and standard deviation from a censored normal sample, Biometrika, 54, (1967), 155-165.
  • Tiku, M. L., Estimating the parameters of log-normal distribution from censored samples, Journal of the American Statistical Association 63(321), (1968), 134--140.
  • Vaughan, D. C., The generalized secant hyperbolic distribution and its properties, Communications in Statistics: Theory and Methods 31, (2002), 219--238.
  • Vaughan, D. C. and Tiku, M. L., Estimation and hypothesis testing for a non-normal bivariate distribution and applications, J. Math. Comput. Modeling 32, (2000), 53--67.
  • Wang, F. K. and Cheng, Y. F., EM algorithm for estimating the Burr XII parameters with multiple censored data, Quality and Reliability Engineering International 26(6), (2010), 615--630.
  • Watkins, A. J., An algorithm for maximum likelihood estimation in the three parameter Burr XII distribution, Computational Statistics and Data Analysis 32, (1999), 19--27.
  • Wingo, D. R., Maximum likelihood methods for fitting the Burr Type XII distribution parameters to life test data, Biometrical Journal 25, (1983), 77-84.
  • Wingo, D. R., Maximum likelihood methods for fitting the Burr Type XII distribution to multiply (progressively) censored life test data, Metrica 40, (1993) 203--210.
  • Yourstone, S. A. and Zimmer, W. J., Non-normality and the Design of Control Charts for Averages, Decision Sciences 23(5), (1992), 1099--1113.
  • Zimmer, W. J. and Burr, I. W., Variables sampling plans based on non-normal populations, Industrial Quality Control 20, (1963), 18--26.

Year 2019, Volume: 68 Issue: 1, 1030 - 1044, 01.02.2019

Fatma Gül Akgül , Şükrü Acıtaş Birdal Şenoğlu

https://doi.org/10.31801/cfsuasmas.501455

Abstract

References

  • Abbasi, B., Hosseinifard, S. Z. and Coit, D. W., A neural network applied to estimate Burr XII distribution parameters, Reliability Engineering and System Safety 95, (2010), 647--654.
  • Abd-Elfattah, A. M., Hassan, A. S. and Nassr, S. G., Estimation in step-stress partially accelerated life tests for the Burr type XII distribution using type I censoring, Statistical Methodology 5, (2008), 502--514.
  • Acitas, S., Kasap, P., Senoglu, B. and Arslan, O., One-step M-estimators: Jones and Faddy's skewed t-distribution, Journal of Applied Statistics 40(7), (2013), 1545--1560.
  • Abuzaid, A. H., The estimation of the Burr-XII parameters with middle-censored data, SpringerPlus 4(101), (2015), 1--10.
  • Afify, A. Z., Cordeiro, G. M., Ortega, E. M. M., Yousof H. M. and Butt, N. S., The four-parameter Burr XII distribution: Properties, regression model and applications, Communications in Statistics-Theory and Methods 47(11), (2018), 2605--2624.
  • Akgul, F. G., and Senoglu, B., Burr XII daglmnn konum ve olcek parametrelerinin tahmini: Monte Carlo Simulasyon Calısması, ISTKON8, Antalya, Turkey.
  • Akgul, F. G., Senoglu, B. and Arslan, T., An alternative distribution to Weibull for modeling the wind speed data: Inverse Weibull distribution, Energy Conversion and Management 114, (2016), 234-240.
  • Arslan, T., Acitas, S., and Senoglu, B., Generalized Lindley and Power Lindley distributions for modeling the wind speed data, Energy Conversion and Management 152, (2017), 300--311.
  • Aydin, D., Akgul, F. G. and Senoglu, B., Robust estimation of the location and the scale parameters of shifted Gompertz distribution, Electronic Journal of Applied Statistical Analysis 11(1), (2018), 92--107.
  • Bowman, K. O. and Shenton, L. R., Weibull distributions when the shape parameter is defined, Computational Statistics and Data Analysis 36, (2001), 299--310.
  • Brano, V. L., Orioli, A., Ciulla, G. and Culotta, S., Quality of wind speed fitting distributions for the urban area of Palermo, Italy, Renewable Energy 36, (2011), 1026--1039.
  • Burr, I. W., Cumulative frequency functions, Annals of Mathematical Statistics, 13, (1942), 215--232.
  • Chou, C. -Y., Chen, C. -H. and Liu, H. -R., Acceptance Control Charts for Non-normal Data, Journal of Applied Statistics 32(1), (2005), 25--36.
  • Dogru, F. Z. and Arslan, O., Optimal B-robust estimators for the parameters of the Burr XII distribution, Journal of Statistical Computation and Simulation 86(6), (2016), 1133--1149.
  • Gomes, A. E., da-Silva, C. Q. and Cordeiro, G. M., Two extended Burr models: Theory and practice, Commun. Stat. Theory - Methods 44, (2015), 1706--1734.
  • Hossain, A. M. and Nath, S. K., Estimation of parameters in the presence of outliers for a burr XII distribution, Communications in Statistics-Theory and Methods 26(3), (1997), 637--652.
  • slam, M. Q. and Tiku, M. L., Multiple linear regression model under nonnormality, Communications in Statistics-Theory and Methods 33(10), (2004), 2443--2467.
  • Kantar, Y. M. and Senoglu, B., A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter, Computers and Geosciences 34(12), (2008), 1900--1909.
  • Klugman, S. A., Panjer, H. H. and Willmot, G. E., Loss Models: From Data to Decisions, Wiley, New York, 1998.
  • Mead, M. E. and Afify, A. Z., On five parameter Burr XII distribution: Properties and applications, South Afr. Stat. J. 51, (2017), 67-80.
  • Nadar, M. and Papadopoulos, A. S., Bayesian analysis for the Burr Type XII distribution based on record values, Statistica 71(4), (2011), 421--435.
  • Panahi, H. and Sayyareh, A. A., Parameter estimation and prediction of order statistics for the Burr Type XII distribution with Type II censoring, Journal of Applied Statistics 41(1), (2014), 215--232.
  • Papalexiou, S. M. and Koutsoyiannis, D., Entropy based derivation of probability distributions: A case study to daily rainfall, Advances in Water Resources 45, (2012) 51--57.
  • Puthenpura, S. and Sinha, N. K., Modified maximum likelihood method for the robust estimation of system parameters from very noisy data, Automatica 22, (1986), 231--235.
  • Rao, G. S., Aslam, M. and Kundu, D., Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength, Communications in Statistics â€" Theory and Methods 44, (2015), 4953--4961.
  • Rodriguez, N., A guide to the Burr type XII distributions, Biometrika 64, (1977), 129--134.
  • Shao, Q., Wong, H., Xia, J. and Ip, W. C., Models for extremes using the extended three- parameter Burr XII system with application to flood frequency analysis, Hydrological Sciences 49, (2004), 685â€"-702.
  • Singh, S. K. and Maddala, G. S., A function for the size distribution of income, Econometrica 44, (1976), 963--970.
  • Soliman, A. A., Estimation of parameters of life from progressively censored data using Burr-XII model, IEEE Transactions on Reliability, 54, (2005), 34--42.
  • Soliman, A. A., Abd Ellah, A. H., Abou-Elheggag, N. A. and Modhesh, A. A., Estimation from Burr type XII distribution using progressive first-failure censored data, Journal of Statistical Computation and Simulation 83(12), (2013), 2270--2290.
  • Şenoğlu, B., Estimating parameters in one-way analysis of covariance model with short-tailed symmetric error distributions, Journal of Computational and Applied Mathematics 201, (2007), 275--283.
  • Swain, J., Venkatraman, S. and Wilson, J., Least squares estimation of distribution function in Johnson’s translation system, Journal of Statistical Computation and Simulation 29, (1988), 271â€"-297.
  • Tadikamalla, P. R., A look at the Burr and related distributions, International Statistical Review 48, (1989), 337--344.
  • Tiku, M. L., Estimating the mean and standard deviation from a censored normal sample, Biometrika, 54, (1967), 155-165.
  • Tiku, M. L., Estimating the parameters of log-normal distribution from censored samples, Journal of the American Statistical Association 63(321), (1968), 134--140.
  • Vaughan, D. C., The generalized secant hyperbolic distribution and its properties, Communications in Statistics: Theory and Methods 31, (2002), 219--238.
  • Vaughan, D. C. and Tiku, M. L., Estimation and hypothesis testing for a non-normal bivariate distribution and applications, J. Math. Comput. Modeling 32, (2000), 53--67.
  • Wang, F. K. and Cheng, Y. F., EM algorithm for estimating the Burr XII parameters with multiple censored data, Quality and Reliability Engineering International 26(6), (2010), 615--630.
  • Watkins, A. J., An algorithm for maximum likelihood estimation in the three parameter Burr XII distribution, Computational Statistics and Data Analysis 32, (1999), 19--27.
  • Wingo, D. R., Maximum likelihood methods for fitting the Burr Type XII distribution parameters to life test data, Biometrical Journal 25, (1983), 77-84.
  • Wingo, D. R., Maximum likelihood methods for fitting the Burr Type XII distribution to multiply (progressively) censored life test data, Metrica 40, (1993) 203--210.
  • Yourstone, S. A. and Zimmer, W. J., Non-normality and the Design of Control Charts for Averages, Decision Sciences 23(5), (1992), 1099--1113.
  • Zimmer, W. J. and Burr, I. W., Variables sampling plans based on non-normal populations, Industrial Quality Control 20, (1963), 18--26.
There are 43 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Fatma Gül Akgül 0000-0001-5034-7596

Şükrü Acıtaş This is me 0000-0002-4131-0086

Birdal Şenoğlu 0000-0003-3707-2393

Publication Date February 1, 2019
Submission Date November 3, 2017
Acceptance Date June 12, 2018
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Akgül, F. G., Acıtaş, Ş., & Şenoğlu, B. (2019). Estimation of the location and the scale parameters of Burr Type XII distribution. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 1030-1044. https://doi.org/10.31801/cfsuasmas.501455
AMA Akgül FG, Acıtaş Ş, Şenoğlu B. Estimation of the location and the scale parameters of Burr Type XII distribution. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):1030-1044. doi:10.31801/cfsuasmas.501455
Chicago Akgül, Fatma Gül, Şükrü Acıtaş, and Birdal Şenoğlu. “Estimation of the Location and the Scale Parameters of Burr Type XII Distribution”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 1030-44. https://doi.org/10.31801/cfsuasmas.501455.
EndNote Akgül FG, Acıtaş Ş, Şenoğlu B (February 1, 2019) Estimation of the location and the scale parameters of Burr Type XII distribution. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 1030–1044.
IEEE F. G. Akgül, Ş. Acıtaş, and B. Şenoğlu, “Estimation of the location and the scale parameters of Burr Type XII distribution”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 1030–1044, 2019, doi: 10.31801/cfsuasmas.501455.
ISNAD Akgül, Fatma Gül et al. “Estimation of the Location and the Scale Parameters of Burr Type XII Distribution”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 1030-1044. https://doi.org/10.31801/cfsuasmas.501455.
JAMA Akgül FG, Acıtaş Ş, Şenoğlu B. Estimation of the location and the scale parameters of Burr Type XII distribution. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1030–1044.
MLA Akgül, Fatma Gül et al. “Estimation of the Location and the Scale Parameters of Burr Type XII Distribution”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 1030-44, doi:10.31801/cfsuasmas.501455.
Vancouver Akgül FG, Acıtaş Ş, Şenoğlu B. Estimation of the location and the scale parameters of Burr Type XII distribution. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):1030-44.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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