Comparison of the robust parameters estimation methods for the two-parameters Lomax distribution. (English) Zbl 1426.62070
Summary: Accurate and precise estimation of parameters in distribution theory is of immense significance. Imprecise and biased estimation of a probability distribution can lead to invalid and erroneous results. In this study, we investigate the Lomax distribution and introduced new four robust point estimation methods such as \(L\)-moments, trimmed \(L\)-moments, probability weighted moments, and generalized probability weighted moments (GPWM). We compare the efficiency of these methods with traditional method of moments based on performance measures such as bias, root-mean-square error and total deviation criteria using simulation study. We concluded that trimmed \(L\)-moments ascertained to be the superior method when the shape parameter is smaller \((q < 3)\) and this assessment is equally valid for larger sample sizes, however, GPWM performs better for higher values of the shape parameter.
MSC:
| 62E15 | Exact distribution theory in statistics |
| 62E10 | Characterization and structure theory of statistical distributions |
| 62F10 | Point estimation |
| 62N05 | Reliability and life testing |
Keywords:
robust estimation; \(L\)-moments; trimmed \(L\)-moments; probability weighted moments; generalized probability weighted moments; Lomax distributionReferences:
| [1] | Abd-Elfattah, A. M.; Alaboud, F. M.; Alharby, A. H., On sample size estimation for Lomax distribution, Australian Journal of Basic and Applied Sciences, 1, 373-378 (2007) |
| [2] | Abd-Elfattah, A. M.; Alharbey, A. H., Estimation of Lomax parameters based on generalized probability weighted moment, Journal of King Saud University - Science, 22, 171-184 (2010) |
| [3] | Abdul-Moniem, I. B.; Abdel-Hameed, H. F., On exponentiated Lomax distribution, International Journal of Mathematical Archive, 3, 2144-2150 (2012) |
| [4] | Abdul-Moniem, I. B.; Seham, M., Transmuted Gompertz distribution, Computational and Applied Mathematics, 1, 88-96 (2015) |
| [5] | Abdul-Moniem, I. B.; Selim, Y. M., TL-moments and L-moments estimation for the generalized Pareto distribution, Applied Mathematical Sciences, 3, 43-52 (2009) · Zbl 1161.62314 |
| [6] | Abu El-Magd, N. A. A., TL-moments of the exponentiated generalized extreme value distribution, Journal of Advanced Research, 1, 351-359 (2010) · doi:10.1016/j.jare.2010.06.003 |
| [7] | Ahmad, A.; Ahmad, S. P.; Ahmed, A., Bayesian analysis of shape parameter of Lomax distribution using different loss functions, International Journal of Statistics and Mathematics, 2, 55-65 (2015) |
| [8] | Ahmad, I.; Abbas, A.; Aslam, M.; Ahmed, I., Total annual rainfall frequency analysis in Pakistan using methods of L-moments and TL-moment, Science International, 27, 2339-2345 (2015) |
| [9] | Ahsanullah, M., Record values of the Lomax distribution, Statistica Neerlandica, 45, 21-29 (1991) · Zbl 0725.62026 · doi:10.1111/j.1467-9574.1991.tb01290.x |
| [10] | Al-Noor, N. H.; Alwan, S. S., Non-bayes and empirical bayes estimators for the reliability and failure rate functions of Lomax distribution, Transactions on Engineering and Sciences, 3, 20-29 (2015) |
| [11] | Al-Noor, N. H.; Alwan, S. S., Non-bayes and and empirical bayes estimators for the shape parameter of Lomax distribution, Mathematical Theory and Modeling, 5, 17-27 (2015) |
| [12] | Ashour, S. K.; Eltehiwy, M. A., Transmuted exponentiated Lomax distribution, Australian Journal of Basic and Applied Sciences, 7, 658-667 (2013) |
| [13] | Asquith, W. H., lmomco: L-moments, trimmed L-moments, L-comoments, and many distributions, R package version 2.1.4 (2015) |
| [14] | Atkinson, A. B.; Harrison, A. J., Distribution of personal wealth in Britain (1978), Cambridge: Cambridge University Press, Cambridge |
| [15] | Attia, A. F.; Shaban, S. A.; Amer, Y. M., Estimation of the bivariate generalized Lomax distribution parameters based on censored samples, International Journal of Contemporary Mathematical Sciences, 9, 257-267 (2014) · doi:10.12988/ijcms.2014.4329 |
| [16] | Attia, A. F.; Shaban, S. A.; Amer, Y. M., Parameter estimation for the bivariate Lomax distribution based on censored samples, Applied Mathematical Sciences, 8, 1711-1721 (2014) · doi:10.12988/ams.2014.42120 |
| [17] | Balakrishnan, N.; Ahsanullah, M., Relations for single and product moments of record values from Lomax distribution, Sankhyā: The Indian Journal of Statistics, Series B, 56, 140-146 (1994) · Zbl 0834.62013 |
| [18] | Batal, I. E. L.; Kareem, A., Statistical properties of Kumaraswamy exponentiated Lomax distribution, Journal of Modern Mathematics and Statistics, 8, 1, 1-7 (2014) |
| [19] | Bílková, D., L-moments and TL-moments as an alternative tool of statistical data analysis, Journal of Applied Mathematics and Physics, 2, 919-929 (2014) · doi:10.4236/jamp.2014.210104 |
| [20] | Bryson, M. C., Heavy-tailed distributions: Properties and tests, Technometrics, 16, 61-68 (1974) · Zbl 0337.62065 · doi:10.1080/00401706.1974.10489150 |
| [21] | Childs, A.; Balakrishnan, N.; Moshref, M., Order statistics from non-identical right-truncated Lomax random variables with applications, Statistical Papers, 42, 187-206 (2001) · Zbl 0998.62049 · doi:10.1007/s003620100050 |
| [22] | Elamir, E. A.; Seheult, A. H., Trimmed L-moments, Computational Statistics & Data Analysis, 43, 299-314 (2003) · Zbl 1220.62019 · doi:10.1016/S0167-9473(02)00250-5 |
| [23] | El-Bassiouny, A. H.; Abdo, N. F.; Shahen, H. S., Exponential Lomax distribution, International Journal of Computer Applications, 121, 24-29 (2015) · doi:10.5120/21602-4713 |
| [24] | Elsherpieny, E.; Hassan, A. S.; El Haroun, N. M., Application of generalized probability weighted moments for skew normal distribution, Asian Journal of Applied Sciences, 2, 1 (2014) |
| [25] | Ghitany, M. E.; Al-Awadhi, F. A.; Alkhalfan, L. A., Marshall-Olkin extended Lomax distribution and its application to censored data., Communications in Statistics—Theory and Methods, 36, 1855-1866 (2007) · Zbl 1122.62081 · doi:10.1080/03610920601126571 |
| [26] | Gomes, M. I.; Guillou, A., Extreme value theory and statistics of univariate extremes: A review, International Statistical Review, 83, 263-292 (2014) · Zbl 07763438 · doi:10.1111/insr.12058 |
| [27] | Greenwood, J. A.; Landwehr, J. M.; Matalas, N. C.; Wallis, J. R., Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form, Water Resources Research, 15, 1049-1054 (1979) · doi:10.1029/WR015i005p01049 |
| [28] | Harris, C. M., The pareto distribution as a queue service discipline, Operations Research, 16, 307-313 (1968) · Zbl 0153.47501 · doi:10.1287/opre.16.2.307 |
| [29] | Hassan, A. S.; Al-Ghamdi, A. S., Optimum step stress accelerated life testing for Lomax distribution, Journal of Applied Sciences Research, 5, 2153-2164 (2009) |
| [30] | Hosking, J. R. M., L-moments: Analysis and estimation of distributions using linear combinations of order statistics, Journal of the Royal Statistical Society. Series B (Methodological), 52, 105-124 (1990) · Zbl 0703.62018 |
| [31] | Hosking, J. R. M., Some theory and practical uses of trimmed L-moments, Journal of Statistical Planning and Inference, 137, 3024-3039 (2007) · Zbl 1331.62255 · doi:10.1016/j.jspi.2006.12.002 |
| [32] | Howlader, H. A.; Hossain, A. M., Bayesian survival estimation of Pareto distribution of the second kind based on failure-censored data, Computational Statistics & Data Analysis, 38, 301-314 (2002) · Zbl 1079.62505 · doi:10.1016/S0167-9473(01)00039-1 |
| [33] | Jamjoom, A.; Alsaiary, Z., TL-moments and L-moments for order statistics from non-identically distributed random variables with applications, Journal of American Science, 9, 102-105 (2013) |
| [34] | Johnson, N.; Kotz, S.; Balakrishnan, N., Continuous univariate distribution, 1 (1994), New York, NY: John Wiley, New York, NY · Zbl 0811.62001 |
| [35] | Landwehr, J. M.; Matalas, N.; Wallis, J., Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles, Water Resources Research, 15, 1055-1064 (1979) · doi:10.1029/WR015i005p01055 |
| [36] | Lemonte, A. J.; Cordeiro, G. M., An extended Lomax distribution, A Journal of Theoretical and Applied Statistics, 47, 800-816 (2013) · Zbl 1440.62062 |
| [37] | Li, X.; Zuo, Y.; Zhuang, X.; Zhu, H., Estimation of fracture trace length distributions using probability weighted moments and L-moments, Engineering Geology, 168, 69-85 (2014) · doi:10.1016/j.enggeo.2013.10.025 |
| [38] | Lingappaiah, G. S., On the pareto distribution of second kind (Lomax distribution), Revista de Matematica e Estatistica, 4, 63-68 (1986) · Zbl 0649.62013 |
| [39] | Ma, Y.; Shi, Y., Inference for Lomax distribution based on type-II progressively hybrid censored data, Journal of Physical Sciences, 17, 33-41 (2013) |
| [40] | Munir, R.; Saleem, M.; Aslam, M.; Ali, S., Comparison of different methods of parameters estimation for Pareto Model, Caspian Journal of Applied Sciences Research, 2, 45-56 (2013) |
| [41] | Myhre, J.; Saunders, S., Screen testing and conditional probability of survival, Lecture Notes-Monograph Series, 166-178 (1982) · doi:10.1214/lnms/1215464847 |
| [42] | Nasiri, P.; Hosseini, S., Statistical inferences for Lomax distribution based on record values (Bayesian and classical), Journal of Modern Applied Statistical Methods, 11, 179-188 (2012) |
| [43] | Naveed-Shahzad, M.; Asghar, Z.; Shehzad, F.; Shahzadi, M., Parameter estimation of power function distribution with TL-moments, Revista Colombiana de Estadística, 38, 321-334 (2015) · Zbl 1437.62074 · doi:10.15446/rce.v38n2.51663 |
| [44] | Nayak, T. K., Multivariate Lomax distribution: Properties and usefulness in reliability theory, Journal of Applied Probability, 24, 170-177 (1987) · Zbl 0615.60082 · doi:10.1017/S0021900200030709 |
| [45] | Rao, R. S.; Durgamamba, A. N.; Kantam, R. R. L., Acceptance sampling plans: Size biased Lomax model, Universal Journal of Applied Mathematics, 2, 176-183 (2014) |
| [46] | Rasmussen, P. F., Generalized probability weighted moments: Application to the generalized Pareto distribution, Water Resources Research, 37, 1745-1751 (2001) · doi:10.1029/2001WR900014 |
| [47] | Shakeel, M.; Haq, M. A.; Hussain, I.; Abdulhamid, A. M.; Faisal, M., Comparison of two new robust parameter estimation methods for the power function distribution, PLOS One, 11, e0160692 (2016) · doi:10.1371/journal.pone.0160692 |
| [48] | Shams, T. M., The Kumaraswamy-generalized Lomax distribution, Middle-East Journal of Scientific Research, 17, 641-646 (2013) |
| [49] | Tahir, M. H.; Cordeiro, G. M.; Mansoor, M.; Zubair, M., The Weibull-Lomax distribution: Properties and applications, Hacettepe Journal of Mathematics and Statistics, 44, 461-480 (2015) · Zbl 1326.60023 |
| [50] | Vidondo, B.; Prairie, Y. T.; Blanco, J. M.; Duarte, C. M., Some aspects of the analysis of size spectra in aquatic ecology, Limnology and Oceanography, 42, 184-192 (1997) · doi:10.4319/lo.1997.42.1.0184 |
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