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Optimal threshold for Pareto tail modelling in the presence of outliers

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  • Safari, Muhammad Aslam Mohd
  • Masseran, Nurulkamal
  • Ibrahim, Kamarulzaman

Abstract

The Pareto distribution is widely applied in many areas of studies such as economics and sciences. An important issues related to Pareto tail modelling is to determine the optimal threshold of the Pareto distribution. One of the methods used for determining the optimal threshold of Pareto distribution is by choosing the threshold that minimizes the goodness-of-fit statistics found based on empirical distribution function (EDF). This study involves determination of the shape parameter of the Pareto distribution using the maximum likelihood method and robust method based on the probability integral transform statistics. In addition, given the particular estimates of the shape parameter, comparison of the performance of several EDF statistics, namely, Kolmogorov–Smirnov, Kuiper, Anderson–Darling, Cramer–von Misses and Watson statistics in determining the optimal threshold in the presence of outliers is studied based on Monte Carlo simulation. Since the EDF statistics are found smallest for Kolmogorov–Smirnov or Kuiper statistics, these two EDF statistics outperformed other EDF statistics considered. The findings are illustrated using a sample of household income data of the Malaysian population. The optimal threshold found can be used to classify the high income earners in Malaysia since Pareto distribution is one of the most frequently used model to describe the upper tail of income distribution.

Suggested Citation

  • Safari, Muhammad Aslam Mohd & Masseran, Nurulkamal & Ibrahim, Kamarulzaman, 2018. "Optimal threshold for Pareto tail modelling in the presence of outliers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 169-180.
    • Handle: RePEc:eee:phsmap:v:509:y:2018:i:c:p:169-180
    DOI: 10.1016/j.physa.2018.06.007
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    as
    1. Danielsson, J. & de Haan, L. & Peng, L. & de Vries, C. G., 2001. "Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 226-248, February.
    2. Cirillo, Pasquale, 2013. "Are your data really Pareto distributed?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(23), pages 5947-5962.
    3. Cirillo, Pasquale & Hüsler, Jürg, 2009. "On the upper tail of Italian firms’ size distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1546-1554.
    4. Jagielski, Maciej & Czyżewski, Kordian & Kutner, Ryszard & Stanley, H. Eugene, 2017. "Income and wealth distribution of the richest Norwegian individuals: An inequality analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 474(C), pages 330-333.
    5. Luckstead, Jeff & Devadoss, Stephen, 2017. "Pareto tails and lognormal body of US cities size distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 465(C), pages 573-578.
    6. Politi, Mauro & Scalas, Enrico, 2008. "Fitting the empirical distribution of intertrade durations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(8), pages 2025-2034.
    7. Soriano-Hernández, P. & del Castillo-Mussot, M. & Córdoba-Rodríguez, O. & Mansilla-Corona, R., 2017. "Non-stationary individual and household income of poor, rich and middle classes in Mexico," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 465(C), pages 403-413.
    8. Clementi, F. & Di Matteo, T. & Gallegati, M., 2006. "The power-law tail exponent of income distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 49-53.
    9. Cowell, Frank A. & Flachaire, Emmanuel, 2007. "Income distribution and inequality measurement: The problem of extreme values," Journal of Econometrics, Elsevier, vol. 141(2), pages 1044-1072, December.
    10. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    11. Didier SORNETTE, 2009. "Dragon-Kings, Black Swans and the Prediction of Crises," Swiss Finance Institute Research Paper Series 09-36, Swiss Finance Institute.
    12. Coronel-Brizio, H.F. & Hernández-Montoya, A.R., 2005. "On fitting the Pareto–Levy distribution to stock market index data: Selecting a suitable cutoff value," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 354(C), pages 437-449.
    13. D. Sornette, "undated". "Dragon-Kings, Black Swans and the Prediction of Crises," Working Papers CCSS-09-005, ETH Zurich, Chair of Systems Design.
    14. Frank A. Cowell, 2008. "Income Distribution and Inequality," Chapters, in: John B. Davis & Wilfred Dolfsma (ed.), The Elgar Companion to Social Economics, chapter 13, Edward Elgar Publishing.
    15. Soo, Kwok Tong, 2005. "Zipf's Law for cities: a cross-country investigation," Regional Science and Urban Economics, Elsevier, vol. 35(3), pages 239-263, May.
    16. Maciej Jagielski & Ryszard Kutner, 2013. "Modeling of income distribution in the European Union with the Fokker-Planck equation," Papers 1301.2076, arXiv.org.
    17. V. I. Yukalov & D. Sornette, 2012. "Statistical Outliers and Dragon-Kings as Bose-Condensed Droplets," Papers 1205.1364, arXiv.org.
    18. Didier Sornette & Guy Ouillon, "undated". "Dragon-kings: Mechanisms, statistical methods and empirical evidence," Working Papers ETH-RC-12-004, ETH Zurich, Chair of Systems Design.
    19. Frank Cowell & Maria-Pia Victoria-Feser, 2007. "Robust stochastic dominance: A semi-parametric approach," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 5(1), pages 21-37, April.
    20. Balakrishnan, P.V. (Sundar) & Miller, James M. & Shankar, S. Gowri, 2008. "Power law and evolutionary trends in stock markets," Economics Letters, Elsevier, vol. 98(2), pages 194-200, February.
    21. Chami Figueira, F. & Moura, N.J. & Ribeiro, M.B., 2011. "The Gompertz–Pareto income distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(4), pages 689-698.
    22. Filimonov, Vladimir & Sornette, Didier, 2015. "Power law scaling and “Dragon-Kings” in distributions of intraday financial drawdowns," Chaos, Solitons & Fractals, Elsevier, vol. 74(C), pages 27-45.
    23. Luckstead, Jeff & Devadoss, Stephen & Danforth, Diana, 2017. "The size distributions of all Indian cities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 474(C), pages 237-249.
    24. Oancea, Bogdan & Andrei, Tudorel & Pirjol, Dan, 2017. "Income inequality in Romania: The exponential-Pareto distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 486-498.
    25. Vladimir Filimonov & Didier Sornette, 2014. "Power law scaling and "Dragon-Kings" in distributions of intraday financial drawdowns," Papers 1407.5037, arXiv.org, revised Apr 2015.
    26. Devadoss, Stephen & Luckstead, Jeff & Danforth, Diana & Akhundjanov, Sherzod, 2016. "The power law distribution for lower tail cities in India," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 442(C), pages 193-196.
    27. Reed, William J., 2001. "The Pareto, Zipf and other power laws," Economics Letters, Elsevier, vol. 74(1), pages 15-19, December.
    28. Rytgaard, Mette, 1990. "Estimation in the Pareto Distribution," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 201-216, November.
    29. Jagielski, Maciej & Kutner, Ryszard, 2013. "Modelling of income distribution in the European Union with the Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(9), pages 2130-2138.
    30. Vladimir FILIMONOV & Didier SORNETTE, 2014. "Power Law Scaling and 'Dragon-Kings' in Distributions of Intraday Financial Drawdowns," Swiss Finance Institute Research Paper Series 14-48, Swiss Finance Institute, revised Apr 2015.
    31. Mark Finkelstein & Howard G. Tucker & Jerry Alan Veeh, 2006. "Pareto Tail Index Estimation Revisited," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(1), pages 1-10.
    32. Andreas Alfons & Matthias Templ & Peter Filzmoser, 2013. "Robust estimation of economic indicators from survey samples based on Pareto tail modelling," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 62(2), pages 271-286, March.
    33. Tomson Ogwang, 2011. "Power laws in top wealth distributions: evidence from Canada," Empirical Economics, Springer, vol. 41(2), pages 473-486, October.
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    2. Muhammad Hilmi Abdul Majid & Kamarulzaman Ibrahim & Nurulkamal Masseran, 2023. "Three-Part Composite Pareto Modelling for Income Distribution in Malaysia," Mathematics, MDPI, vol. 11(13), pages 1-15, June.
    3. Safari, Muhammad Aslam Mohd & Masseran, Nurulkamal & Ibrahim, Kamarulzaman, 2018. "A robust semi-parametric approach for measuring income inequality in Malaysia," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 1-13.
    4. Peipei Ma & Guosheng Li, 2023. "Comparison and Analysis of Detection Methods for Typhoon-Storm Surges Based on Tide-Gauge Data—Taking Coasts of China as Examples," IJERPH, MDPI, vol. 20(4), pages 1-21, February.
    5. Maria Marino & Benedetto Rocchi & Simone Severini, 2021. "Conditional Income Disparity between Farm and Non‐farm Households in the European Union: A Longitudinal Analysis," Journal of Agricultural Economics, Wiley Blackwell, vol. 72(2), pages 589-606, June.
    6. Safari, Muhammad Aslam Mohd & Masseran, Nurulkamal & Ibrahim, Kamarulzaman & Hussain, Saiful Izzuan, 2021. "Measuring income inequality: A robust semi-parametric approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 562(C).
    7. Safari, Muhammad Aslam Mohd & Masseran, Nurulkamal & Ibrahim, Kamarulzaman & Hussain, Saiful Izzuan, 2019. "A robust and efficient estimator for the tail index of inverse Pareto distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 517(C), pages 431-439.

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