An improvement in maximum likelihood estimation of the Burr XII distribution parameters

Ali A. Al-Shomrani; Ali A. Al-Shomrani

doi:10.3934/math.2022961

AIMS Mathematics

2022, Volume 7, Issue 9: 17444-17460. doi: 10.3934/math.2022961

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An improvement in maximum likelihood estimation of the Burr XII distribution parameters

Ali A. Al-Shomrani ^,

Department of Statistics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 03 June 2022 Revised: 18 July 2022 Accepted: 21 July 2022 Published: 28 July 2022
MSC : 60Exx, 62-xx

In this paper, we discuss the parameters estimation of the Burr XII distribution. We know that the most popular method in the literature for parameter estimation is the maximum likelihood method. However, the maximum likelihood estimators (MLEs) are widely known to be biased for small sample sizes. Therefore, this motivate us to obtain approximately unbiased estimators for this distribution' parameters. Precisely, we focus on two bias-correction techniques (analytical and bootstrap approaches) to reduce the biases of the MLEs to the second order of magnitude. In order to compare the performance of these estimators, Monte Carlo simulations are conducted. Lastly, two real-data examples are provided to show the usefulness of these proposed estimators when sample sizes are small.
- Burr XII distribution,
- bias-correction,
- bootstrap
Citation: Ali A. Al-Shomrani. An improvement in maximum likelihood estimation of the Burr XII distribution parameters[J]. AIMS Mathematics, 2022, 7(9): 17444-17460. doi: 10.3934/math.2022961

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Abstract

In this paper, we discuss the parameters estimation of the Burr XII distribution. We know that the most popular method in the literature for parameter estimation is the maximum likelihood method. However, the maximum likelihood estimators (MLEs) are widely known to be biased for small sample sizes. Therefore, this motivate us to obtain approximately unbiased estimators for this distribution' parameters. Precisely, we focus on two bias-correction techniques (analytical and bootstrap approaches) to reduce the biases of the MLEs to the second order of magnitude. In order to compare the performance of these estimators, Monte Carlo simulations are conducted. Lastly, two real-data examples are provided to show the usefulness of these proposed estimators when sample sizes are small.

References

[1]	A. Abd-Elfattah, M. Abu-Moussa, Estimation of stress-strength parameter for Burr type XII distribution based on progressive type-II censoring, ANGLISTICUM J. Assoc.-Inst. English Lang. Am. Stud., 3 (2013), 1857–8187. http://dx.doi.org/10.0001/(aj).v3i0.700 doi: 10.0001/(aj).v3i0.700
[2]	A. H. Abuzaid, The estimation of the Burr-XII parameters with middle-censored data, SpringerPlus, 4 (2015), 101. https://doi.org/10.1186/s40064-015-0856-3 doi: 10.1186/s40064-015-0856-3
[3]	E. K. Al-Hussaini, A characterization of the Burr type XII distribution, Appl. Math. Lett., 4 (1991), 59–61. https://doi.org/10.1016/0893-9659(91)90123-D doi: 10.1016/0893-9659(91)90123-D
[4]	J. Arrué, R. B. Arellano-Valle, H. W. Gómez, Bias reduction of maximum likelihood estimates for a modified skew-normal distribution, J. Stat. Comput. Simul., 86 (2016), 2967–2984. https://doi.org/10.1080/00949655.2016.1143471 doi: 10.1080/00949655.2016.1143471
[5]	F. A. Bhatti, G. G. Hamedani, M. M. A. Sobhi, M. Korkmaz, On the Burr XII-power Cauchy distribution: Properties and applications, AIMS Math., 6 (2021), 7070–7092. https://doi.org/10.3934/math.2021415 doi: 10.3934/math.2021415
[6]	I. W. Burr, Cumulative frequency functions, Ann. Math. Stat., 13 (1942), 215–232. https://doi.org/10.1214/aoms/1177731607 doi: 10.1214/aoms/1177731607
[7]	I. W. Burr, P. J. Cislak, On a general system of distributions: I. Its curve-shape characteristics; II. The sample median, J. Am. Stat. Assoc., 63 (1968), 627–635. https://doi.org/10.2307/2284033 doi: 10.2307/2284033
[8]	J. E. Contreras-Reyes, F. Kahrari, D. D. Cortés, On the modified skew-normal-cauchy distribution: properties, inference and applications, Commun. Stat.-Theor. M., 50 (2021), 3615–3631. https://doi.org/10.1080/03610926.2019.1708942 doi: 10.1080/03610926.2019.1708942
[9]	R. D. Cook, M. E. Johnson, Generalized Burr-Pareto-logistic distributions with applications to a uranium exploration data set, Technometrics, 28 (1986), 123–131. https://doi.org/10.1080/00401706.1986.10488113 doi: 10.1080/00401706.1986.10488113
[10]	G. M. Cordeiro, R. Klein, Bias correction in arma models, Stat. Probabil. Lett., 19 (1994), 169–176. https://doi.org/10.1016/0167-7152(94)90100-7 doi: 10.1016/0167-7152(94)90100-7
[11]	D. R. Cox, E. J. Snell, A general definition of residuals, J. Roy. Stat. Soc. B, 30 (1968), 248–265. https://doi.org/10.1111/j.2517-6161.1968.tb00724.x doi: 10.1111/j.2517-6161.1968.tb00724.x
[12]	B. Efron, Bootstrap methods: Another look at the jackknife, Ann. Stat., 7 (1979), 1–26. https://doi.org/10.1007/978-1-4612-4380-9_41 doi: 10.1007/978-1-4612-4380-9_41
[13]	B. Efron, Computer-intensive methods in statistical regression, SIAM Rev., 30 (1988), 421–449. https://doi.org/10.1137/1030093 doi: 10.1137/1030093
[14]	M. E. Ghitany, S. Al-Awadhi, Maximum likelihood estimation of Burr XII distribution parameters under random censoring, J. Appl. Stat., 29 (2002), 955–965. https://doi.org/10.1080/0266476022000006667 doi: 10.1080/0266476022000006667
[15]	D. E. Giles, Bias reduction for the maximum likelihood estimators of the parameters in the half-logistic distribution, Commun. Stat.-Theor. M., 41 (2012), 212–222. https://doi.org/10.1080/03610926.2010.521278 doi: 10.1080/03610926.2010.521278
[16]	D. E. Giles, H. Feng, R. T. Godwin, On the bias of the maximum likelihood estimator for the two-parameter lomax distribution, Commun. Stat.-Theor. M., 42 (2013), 1934–1950. https://doi.org/10.1080/03610926.2011.600506 doi: 10.1080/03610926.2011.600506
[17]	D. E. Giles, H. Feng, R. T. Godwin, Bias-corrected maximum likelihood estimation of the parameters of the generalized pareto distribution, Commun. Stat.-Theor. M., 45 (2016), 2465–2483. https://doi.org/10.1080/03610926.2014.887104 doi: 10.1080/03610926.2014.887104
[18]	P. H. Johnson, Y. Qi, Y. C. Chueh, Csck mle bias calculation, Working papers, University of Illinois, Urbana-Champaign, Champaign, IL, 2012.
[19]	X. Ling, D. E. Giles, Bias reduction for the maximum likelihood estimator of the parameters of the generalized rayleigh family of distributions, Commun. Stat.-Theor. M., 43 (2014), 1778–1792. https://doi.org/10.1080/03610926.2012.675114 doi: 10.1080/03610926.2012.675114
[20]	J. Mazucheli, A. F. B. Menezes, S. Nadarajah, mle.tools: An R package for maximum likelihood bias correction, R J., 9 (2017), 268–290. https://doi.org/10.32614/RJ-2017-055 doi: 10.32614/RJ-2017-055
[21]	D. Moore, A. S. Papadopoulos, The Burr type XII distribution as a failure model under various loss functions, Microelectron. Reliab., 40 (2000), 2117–2122. https://doi.org/10.1016/S0026-2714(00)00031-7 doi: 10.1016/S0026-2714(00)00031-7
[22]	M. K. Okasha, M. Y. Matter, On the three-parameter Burr type XII distribution and its application to heavy tailed lifetime data, J. Adv. Math., 10 (2015), 3429–3442.
[23]	H. Panahi, Estimation for the parameters of the Burr type XII distribution under doubly censored sample with application to microfluidics data, Int. J. Syst. Assur. Eng., 10 (2019), 510–518. https://doi.org/10.1007/s13198-018-0735-8 doi: 10.1007/s13198-018-0735-8
[24]	A. Prudnikov, Integrals and series, Taylor & Francis, 1986.
[25]	M. K. Rastogi, Y. M. Tripathi, Estimating a parameter of Burr type XII distribution using hybrid censored observations, Int. J. Qual. Reliab. Ma., 28 (2011), 885–893. https://doi.org/10.1108/02656711111162532 doi: 10.1108/02656711111162532
[26]	J. Reath, J. Dong, M. Wang, Improved parameter estimation of the log-logistic distribution with applications, Comput. Stat., 33 (2018), 339–356. https://doi.org/10.1007/s00180-017-0738-y doi: 10.1007/s00180-017-0738-y
[27]	Q. Shao, Notes on maximum likelihood estimation for the three-parameter Burr XII distribution, Comput. Stat. Data An., 45 (2004), 675–687. https://doi.org/10.1016/S0167-9473(02)00367-5 doi: 10.1016/S0167-9473(02)00367-5
[28]	B. D. Stošić, G. M. Cordeiro, Using maple and mathematica to derive bias corrections for two parameter distributions, J. Stat. Comput. Simul., 79 (2009), 751–767. https://doi.org/10.1080/00949650801911047 doi: 10.1080/00949650801911047
[29]	P. R. Tadikamalla, A look at the Burr and related distributions, Int. Stat. Rev./Revue Int. Stat., 48 (1980), 337–344. https://doi.org/10.2307/1402945 doi: 10.2307/1402945
[30]	F. Wang, J. Keats, W. J. Zimmer, Maximum likelihood estimation of the Burr XII parameters with censored and uncensored data, Microelectron. Reliab., 36 (1996), 359–362. https://doi.org/10.1016/0026-2714(95)00077-1 doi: 10.1016/0026-2714(95)00077-1
[31]	M. Wang, W. Wang, Bias-corrected maximum likelihood estimation of the parameters of the weighted lindley distribution, Commun. Stat.-Simul. Comput., 46 (2017), 530–545. https://doi.org/10.1080/03610918.2014.970696 doi: 10.1080/03610918.2014.970696
[32]	A. J. Watkins, An algorithm for maximum likelihood estimation in the three parameter Burr XII distribution, Comput. Stat. Data An., 32 (1999), 19–27. https://doi.org/10.1016/S0167-9473(99)00024-9 doi: 10.1016/S0167-9473(99)00024-9
[33]	D. R. Wingo, Maximum likelihood estimation of Burr type XII distribution parameters under type II censoring, Microelectron. Reliab., 33 (1993), 1251–1257. https://doi.org/10.1016/0026-2714(93)90126-J doi: 10.1016/0026-2714(93)90126-J
[34]	D. R. Wingo, Maximum likelihood methods for fitting the Burr type XII distribution to multiply (progressively) censored life test data, Metrika, 40 (1993), 203–210. https://doi.org/10.1007/BF02613681 doi: 10.1007/BF02613681
[35]	D. Wingo, Maximum likelihood methods for fitting the Burr type XII distribution to life test data, Biometrical J., 25 (1983), 77–84. https://doi.org/10.1002/bimj.19830250109 doi: 10.1002/bimj.19830250109

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