On the Characteristic Function of the Four-Parameter Generalized Beta of the Second Kind (GB2) Distribution and Its Approximation to the Singh-Maddala, Dagum, and Fisk Distributions
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Keywords:
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Generalized Beta of the Second Kind, Characteristic Function, Moment, Gamma Function, Beta Function, Mclaurin Series
Abstract
Researchers have thoroughly investigated generalized distributions due to their inherent flexibility, which allows them to include several well-known distributions as special cases. Among these, the four parameter Generalized Beta of the Second Kind (GB2) distribution stands out as one of the most versatile frameworks in probability theory. Despite its broad applications, the GB2 distribution’s characteristic function, a critical tool in probability and statistical analysis, lacks a closed-form solution in the existing literature. This study pursues two primary objectives: first, to derive the characteristic function and the kth moment of the GB2 distribution, and second, to demonstrate how the GB2 distribution can serve as a close approximation to the Singh-Maddala, Dagum, and Fisk distributions using its characteristic function and kth moment. These derivations and approximations rely on gamma and beta functions, supplemented by the Maclaurin series expansion.
References
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Ahsanullah, M. and A. Alzaatreh (2018). Some Characterizations of the Log-Logistic Distribution. Stochastics and Quality Control, 33(1); 23–29.
Chotikapanich, D., W. E. Griffiths, G. Hajargasht, W. Karunarathne, and D. S. Prasada Rao (2018). Using the GB2 Income Distribution. Econometrics, 6(2); 21.
Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett (1990). Applications of the GB2 Family of Distributions in Modeling Insurance Loss Processes. Insurance Mathematics and Economics, 9(4); 257–272.
Dey, S., B. Al-Zahrani, and S. Basloom (2017). Dagum Distribution: Properties and Different Methods of Estimation. International Journal of Statistics and Probability, 6(2); 74.
Domma, F. and P. F. Perri (2009). Some Developments on the Log-Dagum Distribution. Statistical Methods and Applications, 18(2); 205–220.
Feng, S., R. V. Burkhauser, and J. S. Butler (2006). Levels and Long-Term Trends in Earnings Inequality: Overcoming Current Population Survey Censoring Problems Using the GB2 Distribution. Journal of Business and Economic Statistics, 24(1); 57–62.
Graf, M. and D. Nedyalkova (2014). Modeling of Income and Indicators of Poverty and Social Exclusion Using the Generalized Beta Distribution of the Second Kind. Review of Income and Wealth, 60(4); 821–842.
Han, X. and Y. Cheng (2019). Does the “Missing” High-Income Matter? - Income Distribution and Inequality Revisited with Truncated Distribution. China Economic Review, 57; 101337.
Jenkins, S. P. (2009). Distributionally-Sensitive Inequality Indices and the GB2 Income Distribution. Review of Income and Wealth, 55(2); 392–398.
Jenkins, S. P. (2011). Inequality and the GB2 Income Distribution. SSRN Electronic Journal; 2831.
Kendall, M. G. and A. Stuart (1968). The Advanced Theory of Statistics. The Statistician, 18(2); 163.
Kleiber, C. (2008). A Guide to the Dagum Distributions. In Modeling Income Distributions and Lorenz Curves. Society of Actuaries, pages 97–117.
Klugman, S. A., H. H. Panjer, and G. E. Willmot (2012). Loss Models: From Data to Decisions. Third edition.
Kurniasari, D., Y. Anggun Anggrainy, and M. Usman (2017). Simulation of Moment, Cumulant, Kurtosis and the Characteristics Function of Dagum Distribution. IJCSNS International Journal of Computer Science and Network Security, 17(8); 264–270.
Lukacs, E. (1963). Applications of Characteristic Functions in Statistics. Sankhya: The Indian Journal of Statistics, Series A (1961-2002), 25(2); 175–188.
McDonald, J. B. (1984). Some Generalized Functions for the Size Distribution of Income. Econometrica, 52(3); 647.
McDonald, J. B., J. Sorensen, and P. A. Turley (2013). Skewness and Kurtosis Properties of Income Distribution Models. Review of Income and Wealth, 59(2); 360–374.
McDonald, J. B. and Y. J. Xu (1995). A Generalization of the Beta Distribution with Applications. Journal of Econometrics, 66(1–2); 133–152.
Parker, S. C. (1999). The Generalized Beta as a Model for the Distribution of Earnings. Economics Letters, 62(2); 197–200.
Pérez, C. G. and M. P. Alaiz (2011). Using the Dagum Model to Explain Changes in Personal Income Distribution. Applied Economics, 43(28); 4377–4386.
Singh, S. K. and G. S. Maddala (1976). A Function for Size Distribution of Incomes. Econometrica, 44(5); 963.
Slottje, D. (2010). Modeling Income Distributions and Lorenz Curves. The Journal of Economic Inequality, 8(4); 525–526.
Warsono (2010). Remarks on Moment Properties of Generalized Distributions. In Proceedings of the Third International Conference on Mathematics and Natural Sciences (ICMNS 2010). Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung, pages 1123–1131.
Warsono, D. Kurniasari, Widiarti, M. Usman, and F. A. M. Elfaki (2017). The Moment Generating Function of the Four-Parameter Generalized F Distribution and Related Generalized Distributions. Global Journal of Pure and Applied Mathematics, 13(3); 941–956.
Warsono, Waryoto, M. Usman, and Wamiliana (2018). Estimation of Parameters of Generalized Beta of the Second Kind (GB2) Distribution by Maximum Likelihood Estimation (MLE) and Newton-Raphson Iteration. Far East Journal of Mathematical Sciences (FJMS), 105(1); 1–20.
Ahsanullah, M. and A. Alzaatreh (2018). Some Characterizations of the Log-Logistic Distribution. Stochastics and Quality Control, 33(1); 23–29.
Chotikapanich, D., W. E. Griffiths, G. Hajargasht, W. Karunarathne, and D. S. Prasada Rao (2018). Using the GB2 Income Distribution. Econometrics, 6(2); 21.
Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett (1990). Applications of the GB2 Family of Distributions in Modeling Insurance Loss Processes. Insurance Mathematics and Economics, 9(4); 257–272.
Dey, S., B. Al-Zahrani, and S. Basloom (2017). Dagum Distribution: Properties and Different Methods of Estimation. International Journal of Statistics and Probability, 6(2); 74.
Domma, F. and P. F. Perri (2009). Some Developments on the Log-Dagum Distribution. Statistical Methods and Applications, 18(2); 205–220.
Feng, S., R. V. Burkhauser, and J. S. Butler (2006). Levels and Long-Term Trends in Earnings Inequality: Overcoming Current Population Survey Censoring Problems Using the GB2 Distribution. Journal of Business and Economic Statistics, 24(1); 57–62.
Graf, M. and D. Nedyalkova (2014). Modeling of Income and Indicators of Poverty and Social Exclusion Using the Generalized Beta Distribution of the Second Kind. Review of Income and Wealth, 60(4); 821–842.
Han, X. and Y. Cheng (2019). Does the “Missing” High-Income Matter? - Income Distribution and Inequality Revisited with Truncated Distribution. China Economic Review, 57; 101337.
Jenkins, S. P. (2009). Distributionally-Sensitive Inequality Indices and the GB2 Income Distribution. Review of Income and Wealth, 55(2); 392–398.
Jenkins, S. P. (2011). Inequality and the GB2 Income Distribution. SSRN Electronic Journal; 2831.
Kendall, M. G. and A. Stuart (1968). The Advanced Theory of Statistics. The Statistician, 18(2); 163.
Kleiber, C. (2008). A Guide to the Dagum Distributions. In Modeling Income Distributions and Lorenz Curves. Society of Actuaries, pages 97–117.
Klugman, S. A., H. H. Panjer, and G. E. Willmot (2012). Loss Models: From Data to Decisions. Third edition.
Kurniasari, D., Y. Anggun Anggrainy, and M. Usman (2017). Simulation of Moment, Cumulant, Kurtosis and the Characteristics Function of Dagum Distribution. IJCSNS International Journal of Computer Science and Network Security, 17(8); 264–270.
Lukacs, E. (1963). Applications of Characteristic Functions in Statistics. Sankhya: The Indian Journal of Statistics, Series A (1961-2002), 25(2); 175–188.
McDonald, J. B. (1984). Some Generalized Functions for the Size Distribution of Income. Econometrica, 52(3); 647.
McDonald, J. B., J. Sorensen, and P. A. Turley (2013). Skewness and Kurtosis Properties of Income Distribution Models. Review of Income and Wealth, 59(2); 360–374.
McDonald, J. B. and Y. J. Xu (1995). A Generalization of the Beta Distribution with Applications. Journal of Econometrics, 66(1–2); 133–152.
Parker, S. C. (1999). The Generalized Beta as a Model for the Distribution of Earnings. Economics Letters, 62(2); 197–200.
Pérez, C. G. and M. P. Alaiz (2011). Using the Dagum Model to Explain Changes in Personal Income Distribution. Applied Economics, 43(28); 4377–4386.
Singh, S. K. and G. S. Maddala (1976). A Function for Size Distribution of Incomes. Econometrica, 44(5); 963.
Slottje, D. (2010). Modeling Income Distributions and Lorenz Curves. The Journal of Economic Inequality, 8(4); 525–526.
Warsono (2010). Remarks on Moment Properties of Generalized Distributions. In Proceedings of the Third International Conference on Mathematics and Natural Sciences (ICMNS 2010). Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung, pages 1123–1131.
Warsono, D. Kurniasari, Widiarti, M. Usman, and F. A. M. Elfaki (2017). The Moment Generating Function of the Four-Parameter Generalized F Distribution and Related Generalized Distributions. Global Journal of Pure and Applied Mathematics, 13(3); 941–956.
Warsono, Waryoto, M. Usman, and Wamiliana (2018). Estimation of Parameters of Generalized Beta of the Second Kind (GB2) Distribution by Maximum Likelihood Estimation (MLE) and Newton-Raphson Iteration. Far East Journal of Mathematical Sciences (FJMS), 105(1); 1–20.
Authors
Warsono, & Kurniasari, D. (2025). On the Characteristic Function of the Four-Parameter Generalized Beta of the Second Kind (GB2) Distribution and Its Approximation to the Singh-Maddala, Dagum, and Fisk Distributions. Science and Technology Indonesia, 10(1), 201–211. https://doi.org/10.26554/sti.2025.10.1.201-211
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