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Year 2022, Volume: 51 Issue: 4, 1189 - 1210, 01.08.2022
https://doi.org/10.15672/hujms.920545

Abstract

References

  • [1] A. Andreev , A. Kanto and P. Malo, Computational examples of a new method for distribution selection in the Pearson system, J. Appl. Stat. 34 (4), 487-506, 2007.
  • [2] K. Arora and V.P. Singh, A comparative evaluation of the estimators of the log Pearson type (LP) 3 distribution, J. Hydrol. 105 (1-2), 19-37, 1989.
  • [3] F. Ashkar and B. Bobée, The generalized method of moments as applied to problems of flood frequency analysis: some practical results for the log-Pearson type 3 distribution, J. Hydrol. 90 (3-4), 199-217, 1987.
  • [4] K.O. Bowman and L.R. Shenton, Approximate percentage points for Pearson distributions, Biometrika 66 (1), 147-151, 1979.
  • [5] K.O. Bowman and L.R. Shenton, Notes on the distribution of $\sqrt b_1$ in sampling from pearson distributions, Biometrika 60 (1), 155-167, 1973.
  • [6] Y. Büyükkör and A.K. Şehirlioğlu, A Robust Regression Method Based on Pearson Type VI Distribution, in: Advances in Econometrics, Operational Research, Data Science and Actuarial Studies, 117-142, Switzerland:Springer, 2022.
  • [7] S. Chen and H. Nie, Lognormal sum approximation with a variant of type IV Pearson distribution, IEEE Commun. Lett. 12 (9), 630-632, 2008.
  • [8] A.C. Cohen, Estimation of parameters in truncated Pearson frequency distributions, Ann. Math. Stat. 22 (2), 256-265, 1951.
  • [9] H. Cramér, Mathematical Methods of Statistics, NJ: Princeton University Press, 1946.
  • [10] K.A. Dunning and J.N. Hanson, Generalized Pearson distributions and nonlinear programing, J. Stat. Comput. Simul. 6 (2), 115-128, 1977.
  • [11] W.P. Elderton and N.L. Johnson, System of Frequency Curves, USA: Cambridge University Press, 1969.
  • [12] J.F. England, J.D. Salas and R.D. Jarrett, Comparisons of two moments-based estimators that utilize historical and paleoflood data for the log Pearson type III distribution, Water Resour. Res. 39 (9), 1-16, 2003.
  • [13] A.M. Fiori and M. Zenga, The meaning of kurtosis, the influence function and an early intuition by L. Faleschini, Statistica 65 (2), 135-144, 2005.
  • [14] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics, The Approach Based on Influence Functions, New York: Wiley, 1986.
  • [15] F.R. Hampel, The influence curve and its role in robust estimation, J. Amer. Statist. Assoc. 69 (346), 383-393, 1974.
  • [16] P.J. Huber and E.M. Ronchetti, Robust Statistics, New York: John Wiley and Sons, 2009.
  • [17] T. Isogai, On using influence functions for testing multivariate normality, J. Amer. Statist. Assoc. 41 (1), 169-186, 1989.
  • [18] M.G. Kendall, A. Stuart, J.K. Ord, S.F. Arnold, A. O’Hagan and J. Forster, Kendall’s Advanced Theory of Statistics (Vol.1), London: Griffin, 1987.
  • [19] I.A. Koutrouvelis and G.C. Canavos, Estimation in the Pearson type 3 distribution, Water Resour. Res. 35 (9), 2693-2704, 1999.
  • [20] R.V. Krejcie and D.W. Morgan, Determining sample size for research activities, Educ. Psychol. Meas. 30 (3), 607-610, 1970.
  • [21] B. Lahcene, On Pearson families of distributions and its applications, Afr. J. Math. Comput. Sci. Res. 6 (5), 108-117, 2013.
  • [22] K.W. Liao and N.I.D.R. Biton, A heuristic optimization considering probabilistic constraints via an equivalent single variable Pearson distribution system, Appl. Soft Comput. 2019 (78), 670-684, 2019.
  • [23] N.C. Matalas and J.R. Wallis, Eureka! It fits a Pearson type: 3 distribution, Water Resour. Res. 9 (2), 281-298, 1973.
  • [24] Y. Nagahara, A method of simulating multivariate nonnormal distributions by the Pearson distribution system and estimation, Comput Stat Data Anal 40 (2004), 1-29, 2004.
  • [25] Y. Nagahara, Non-Gaussian filter and smoother based on the Pearson distribution system, J. Time Ser. Anal. 24 (6), 721-738, 2003.
  • [26] Y. Nagahara, The PDF and CF of Pearson type IV distributions and the ML estimation of the parameters, Stat Probab Lett 43 (3), 251-264, 1999.
  • [27] B. Naghavi, J. Cruise and K. Arora A comparative evaluation of three estimators of log Pearson type 3 distribution, Transp. Res. Rec. 1279, 103-112, 1990.
  • [28] N.U. Nair and P.G. Sankaran, Characterization of the Pearson family of distributions, IEEE Trans Reliab 40 (1), 75-77, 1991.
  • [29] O. Özdemir and H. Emeç, CDS Primleri, Hisse Senedi Piyasası ve Petrol Piyasası Arasındaki Oynaklık Yayılımı in Ekonometride Ampirik Çalışmalar, 137-160, Ankara: Nobel, 2020.
  • [30] R.S. Parrish, On an integrated approach to member selection and parameter estimation for Pearson distributions, Comput Stat Data Anal 1, 239-255, 1983.
  • [31] H.O. Posten, The robustness of the one-sample t-test over the Pearson system, J. Stat. Comput. Simul. 9 (2), 133-149, 1979.
  • [32] M.H. Quenouille, Notes on bias in estimation, Biometrika 43 (3/4), 353-360, 1956.
  • [33] R. Serfling,Asymptotic Relative Efficiency in Estimation, International Encyclopedia Of Statistical Science, 68-72, 2011.
  • [34] M. Shakil, B.M.G. Kibria and J.N. Singh, A new family of distributions based on the generalized Pearson differential equation with some applications, Austrian J. Stat. 39 (3), 259278, 2010.
  • [35] M. Shauly and Y. Parmet, Comparison of Pearson distribution system and response modeling methodology (RMM) as models for process capability analysis of skewed data, Qual. Reliab. Eng. Int. 2011 (27), 681-687, 2011.
  • [36] V.P. Singh and K. Singh, Parameter estimation for log-Pearson type III distribution by POME, J Hydraul Eng 114 (1), 112-122, 1988.
  • [37] H. Solomon and M.A. Stephens, Approximations to density functions using Pearson curves, J. Amer. Statist. Assoc. 73 (361), 153-160, 1978.
  • [38] S. Stavroyiannis, On the generalised Pearson distribution for application in financial time series modelling, Glob. Bus. Econ. Rev 16 (1), 1-14, 2014.
  • [39] J. Sun, A. Kabán and J.M. Garibaldi, Robust mixture clustering using Pearson type VII distribution, Pattern Recognit. Lett. 31 (2010), 2447-2454, 2010.
  • [40] A.K. Şehirlioğlu and S. Dündar, Pearson Dağılış Ailesi,İzmir: Ege Üniversitesi Basımevi, 2014.
  • [41] M. Ünlü, Comparison of The Parameter Estimators of The Main Types of Pearson Distributions by Robustness Criteria, PhD Thesis, Dokuz Eylül University, 2019.
  • [42] R. Willink, A closed-form expression for the Pearson type IV distribution function, Aust N Z J Stat 50 (2), 199-205, 2008.
  • [43] Z. Winiewski, M-estimation with probabilistic models of geodetic observations, J. Geod. 88 (10), 941-957, 2014.

Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria

Year 2022, Volume: 51 Issue: 4, 1189 - 1210, 01.08.2022
https://doi.org/10.15672/hujms.920545

Abstract

Pearson's differential equation is used for fitting a distribution to a data set. The differential equation has some alternative moment-based estimators (depending on the transformation to data). The estimator used when no transformation is made on the data set has 4 elements, and the estimators that require any transformation have 3 elements. We describe all elements of the estimators by corresponding vectors. One of the factors affecting the preference of an estimator is robustness. We use covariance matrix, bias, relative efficiency and influence function as our robustness criteria. Our aim is to compare the performance of the estimators of the differential equation for some specific distributions (namely Type I, Type IV, Type VI and Type III). 10,000 samples with specific sizes were selected with replacement. Also, we evaluated the performance of the estimators over real-life data. Considering the results, there is no best estimator in all criteria. Depending on the criterion to be based, the estimator to be preferred varies.

References

  • [1] A. Andreev , A. Kanto and P. Malo, Computational examples of a new method for distribution selection in the Pearson system, J. Appl. Stat. 34 (4), 487-506, 2007.
  • [2] K. Arora and V.P. Singh, A comparative evaluation of the estimators of the log Pearson type (LP) 3 distribution, J. Hydrol. 105 (1-2), 19-37, 1989.
  • [3] F. Ashkar and B. Bobée, The generalized method of moments as applied to problems of flood frequency analysis: some practical results for the log-Pearson type 3 distribution, J. Hydrol. 90 (3-4), 199-217, 1987.
  • [4] K.O. Bowman and L.R. Shenton, Approximate percentage points for Pearson distributions, Biometrika 66 (1), 147-151, 1979.
  • [5] K.O. Bowman and L.R. Shenton, Notes on the distribution of $\sqrt b_1$ in sampling from pearson distributions, Biometrika 60 (1), 155-167, 1973.
  • [6] Y. Büyükkör and A.K. Şehirlioğlu, A Robust Regression Method Based on Pearson Type VI Distribution, in: Advances in Econometrics, Operational Research, Data Science and Actuarial Studies, 117-142, Switzerland:Springer, 2022.
  • [7] S. Chen and H. Nie, Lognormal sum approximation with a variant of type IV Pearson distribution, IEEE Commun. Lett. 12 (9), 630-632, 2008.
  • [8] A.C. Cohen, Estimation of parameters in truncated Pearson frequency distributions, Ann. Math. Stat. 22 (2), 256-265, 1951.
  • [9] H. Cramér, Mathematical Methods of Statistics, NJ: Princeton University Press, 1946.
  • [10] K.A. Dunning and J.N. Hanson, Generalized Pearson distributions and nonlinear programing, J. Stat. Comput. Simul. 6 (2), 115-128, 1977.
  • [11] W.P. Elderton and N.L. Johnson, System of Frequency Curves, USA: Cambridge University Press, 1969.
  • [12] J.F. England, J.D. Salas and R.D. Jarrett, Comparisons of two moments-based estimators that utilize historical and paleoflood data for the log Pearson type III distribution, Water Resour. Res. 39 (9), 1-16, 2003.
  • [13] A.M. Fiori and M. Zenga, The meaning of kurtosis, the influence function and an early intuition by L. Faleschini, Statistica 65 (2), 135-144, 2005.
  • [14] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics, The Approach Based on Influence Functions, New York: Wiley, 1986.
  • [15] F.R. Hampel, The influence curve and its role in robust estimation, J. Amer. Statist. Assoc. 69 (346), 383-393, 1974.
  • [16] P.J. Huber and E.M. Ronchetti, Robust Statistics, New York: John Wiley and Sons, 2009.
  • [17] T. Isogai, On using influence functions for testing multivariate normality, J. Amer. Statist. Assoc. 41 (1), 169-186, 1989.
  • [18] M.G. Kendall, A. Stuart, J.K. Ord, S.F. Arnold, A. O’Hagan and J. Forster, Kendall’s Advanced Theory of Statistics (Vol.1), London: Griffin, 1987.
  • [19] I.A. Koutrouvelis and G.C. Canavos, Estimation in the Pearson type 3 distribution, Water Resour. Res. 35 (9), 2693-2704, 1999.
  • [20] R.V. Krejcie and D.W. Morgan, Determining sample size for research activities, Educ. Psychol. Meas. 30 (3), 607-610, 1970.
  • [21] B. Lahcene, On Pearson families of distributions and its applications, Afr. J. Math. Comput. Sci. Res. 6 (5), 108-117, 2013.
  • [22] K.W. Liao and N.I.D.R. Biton, A heuristic optimization considering probabilistic constraints via an equivalent single variable Pearson distribution system, Appl. Soft Comput. 2019 (78), 670-684, 2019.
  • [23] N.C. Matalas and J.R. Wallis, Eureka! It fits a Pearson type: 3 distribution, Water Resour. Res. 9 (2), 281-298, 1973.
  • [24] Y. Nagahara, A method of simulating multivariate nonnormal distributions by the Pearson distribution system and estimation, Comput Stat Data Anal 40 (2004), 1-29, 2004.
  • [25] Y. Nagahara, Non-Gaussian filter and smoother based on the Pearson distribution system, J. Time Ser. Anal. 24 (6), 721-738, 2003.
  • [26] Y. Nagahara, The PDF and CF of Pearson type IV distributions and the ML estimation of the parameters, Stat Probab Lett 43 (3), 251-264, 1999.
  • [27] B. Naghavi, J. Cruise and K. Arora A comparative evaluation of three estimators of log Pearson type 3 distribution, Transp. Res. Rec. 1279, 103-112, 1990.
  • [28] N.U. Nair and P.G. Sankaran, Characterization of the Pearson family of distributions, IEEE Trans Reliab 40 (1), 75-77, 1991.
  • [29] O. Özdemir and H. Emeç, CDS Primleri, Hisse Senedi Piyasası ve Petrol Piyasası Arasındaki Oynaklık Yayılımı in Ekonometride Ampirik Çalışmalar, 137-160, Ankara: Nobel, 2020.
  • [30] R.S. Parrish, On an integrated approach to member selection and parameter estimation for Pearson distributions, Comput Stat Data Anal 1, 239-255, 1983.
  • [31] H.O. Posten, The robustness of the one-sample t-test over the Pearson system, J. Stat. Comput. Simul. 9 (2), 133-149, 1979.
  • [32] M.H. Quenouille, Notes on bias in estimation, Biometrika 43 (3/4), 353-360, 1956.
  • [33] R. Serfling,Asymptotic Relative Efficiency in Estimation, International Encyclopedia Of Statistical Science, 68-72, 2011.
  • [34] M. Shakil, B.M.G. Kibria and J.N. Singh, A new family of distributions based on the generalized Pearson differential equation with some applications, Austrian J. Stat. 39 (3), 259278, 2010.
  • [35] M. Shauly and Y. Parmet, Comparison of Pearson distribution system and response modeling methodology (RMM) as models for process capability analysis of skewed data, Qual. Reliab. Eng. Int. 2011 (27), 681-687, 2011.
  • [36] V.P. Singh and K. Singh, Parameter estimation for log-Pearson type III distribution by POME, J Hydraul Eng 114 (1), 112-122, 1988.
  • [37] H. Solomon and M.A. Stephens, Approximations to density functions using Pearson curves, J. Amer. Statist. Assoc. 73 (361), 153-160, 1978.
  • [38] S. Stavroyiannis, On the generalised Pearson distribution for application in financial time series modelling, Glob. Bus. Econ. Rev 16 (1), 1-14, 2014.
  • [39] J. Sun, A. Kabán and J.M. Garibaldi, Robust mixture clustering using Pearson type VII distribution, Pattern Recognit. Lett. 31 (2010), 2447-2454, 2010.
  • [40] A.K. Şehirlioğlu and S. Dündar, Pearson Dağılış Ailesi,İzmir: Ege Üniversitesi Basımevi, 2014.
  • [41] M. Ünlü, Comparison of The Parameter Estimators of The Main Types of Pearson Distributions by Robustness Criteria, PhD Thesis, Dokuz Eylül University, 2019.
  • [42] R. Willink, A closed-form expression for the Pearson type IV distribution function, Aust N Z J Stat 50 (2), 199-205, 2008.
  • [43] Z. Winiewski, M-estimation with probabilistic models of geodetic observations, J. Geod. 88 (10), 941-957, 2014.
There are 43 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Mustafa Ünlü 0000-0001-6652-8535

Ali Şehirlioğlu 0000-0001-5190-6740

Publication Date August 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 4

Cite

APA Ünlü, M., & Şehirlioğlu, A. (2022). Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria. Hacettepe Journal of Mathematics and Statistics, 51(4), 1189-1210. https://doi.org/10.15672/hujms.920545
AMA Ünlü M, Şehirlioğlu A. Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria. Hacettepe Journal of Mathematics and Statistics. August 2022;51(4):1189-1210. doi:10.15672/hujms.920545
Chicago Ünlü, Mustafa, and Ali Şehirlioğlu. “Comparison of the Alternative Parameter Estimators of Pearson Distributions by Robustness Criteria”. Hacettepe Journal of Mathematics and Statistics 51, no. 4 (August 2022): 1189-1210. https://doi.org/10.15672/hujms.920545.
EndNote Ünlü M, Şehirlioğlu A (August 1, 2022) Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria. Hacettepe Journal of Mathematics and Statistics 51 4 1189–1210.
IEEE M. Ünlü and A. Şehirlioğlu, “Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 4, pp. 1189–1210, 2022, doi: 10.15672/hujms.920545.
ISNAD Ünlü, Mustafa - Şehirlioğlu, Ali. “Comparison of the Alternative Parameter Estimators of Pearson Distributions by Robustness Criteria”. Hacettepe Journal of Mathematics and Statistics 51/4 (August 2022), 1189-1210. https://doi.org/10.15672/hujms.920545.
JAMA Ünlü M, Şehirlioğlu A. Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria. Hacettepe Journal of Mathematics and Statistics. 2022;51:1189–1210.
MLA Ünlü, Mustafa and Ali Şehirlioğlu. “Comparison of the Alternative Parameter Estimators of Pearson Distributions by Robustness Criteria”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 4, 2022, pp. 1189-10, doi:10.15672/hujms.920545.
Vancouver Ünlü M, Şehirlioğlu A. Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria. Hacettepe Journal of Mathematics and Statistics. 2022;51(4):1189-210.