Testing Special Cases of the GB2 Distribution
DOI:
https://doi.org/10.25071/1874-6322.40613Keywords:
Wald test, Likelihood ratio test, nonmonotonic power, log-transformed parametersAbstract
We examine the power of hypothesis tests leading to four widely used special cases of the four-parameter generalized beta distribution of the second kind (GB2). These are the Singh-Maddala, Dagum, beta-2 and Fisk distributions. For the Singh-Maddala, Dagum, and Fisk distributions, the power of Wald tests is nonmonotonic when the true parameter values are greater than those in the null hypothesis. As the difference between the hypothesized values of the parameters and their actual values increases, the power increases and then declines. Wald tests on log-transformations of the parameters are more powerful than those on the original parameters; their power functions are also nonmonotonic but the decline in power begins at higher values of the parameters. The likelihood ratio and Lagrange multiplier tests have power functions that increase monotonically for true parameter values beyond the hypothesized values, and are more powerful than their Wald counterparts, but only marginally so relative to the Wald test on the log-transformed parameters. For parameter values less than the hypothesized values there are no large differences in the power of all four tests. Tests for the remaining distribution, the beta-2 distribution, have power functions that are well behaved and exhibit only small differences. We provide an extensive online supplement containing several theoretical results as well as further insights into the power behavior of the tests.
References
Bickel, P.J., and K.A. Doksum 2015 Mathematical Statistics, Basic Ideas and Selected Topics, Volume 1, Second edition Boca Raton, FL: CRC Press.
Butler, R.J., and J.B. McDonald 1989 “Using Incomplete Moments to Measure Inequality”, Journal of Econometrics 42: 109–120. doi.org/10.1016/0304-4076(89)90079-1
Cao, M., W. Sun, and M.R. Kosorok 2013 “The Optimal Power Puzzle: Scrutiny of the Momnotone Likelihood Ratio Assumption in Multiple Testing”, Biometrika 100(2): 495–502. doi.org/10.1093/biomet/ast001
Chotikapanich, D., and W.E. Griffiths 2023 “Poverty in Southeast Asia”, Ch. 65 in J. Silber (Eds.) Handbook of Research on Measuring Poverty and Deprivation Edward Elgar Publishing. pp. 696–709.
Chotikapanich, D., W.E. Griffiths, G. Hajargasht, W. Karunarathne, and D.S.P. Rao 2018 “Using the GB2 Income Distribution”, Econometrics, 6(2): 21, 1–24. doi.org/10.3390/econometrics6020021
Crainiceanu, C.M., and T.J. Vogelsang 2007 “Nonmonotonic Power for Tests of a Mean Shift in a Time Series”, Journal of Statistical Computation and Simulation, 77: 457–476. doi.org/10.1080/10629360600569394
Dagum, C. 1977 “A New Model of Personal Income Distribution: Specification and Estimation”, Economie Appliquée, 30: 413–437. doi.org/10.3406/ecoap.1977.4213
Fisk, P.R. 1961 “The graduation of income distributions”, Econometrica, 29(2): 171–185.
Greene, W.H. 2018 Econometric Analysis (8th ed.) Upper Saddle River, NJ: Pearson.
Griffiths, W.E., and R.C. Hill 2022 “On the Power of the F-test for Hypotheses in a Linear Model”, The American Statistician, 76(1): 78–84. doi.org/10.1080/00031305.2021.1979652
Jenkins, S.P. 2009 “Distributionally-Sensitive Inequality Indices and the GB2 Income Distribution”, Review of Income and Wealth, 55(2): 392–398. doi.org/10.1111/j.1475-4991.2009.00318.x
Jenkins, S.P. 2014 “GB2LFIT: Stata module to fit Generalized Beta of the Second Kind distribution by maximum likelihood (log parameter metric)”, Statistical Software Components S457897, Boston College Department of Economics.
Kleiber, C. 2008 “A Guide to Dagum Distributions”, in D. Chotikapanich (Ed.) Modeling Income Distributions and Lorenz Curves New York: Springer. pp. 97–118. doi.org/10.1007/978-0-387-72796-7_6
Kleiber, C., and S. Kotz 2003 Statistical Size Distributions in Economics and Actuarial Sciences New York: John Wiley and & Sons.
McDonald, J.B. 1984 “Some generalized functions for the size distribution of income”, Econometrica, 52(3): 647–663.
McDonald, J.B., and M.R. Ransom 2008 “The Generalized Beta Distribution as a Model for the Distribution of Income: Estimation of Related Measures of Inequality”, in D. Chotikapanich (Ed.) Modeling Income Distributions and Lorenz Curves New York: Springer. pp. 147–166. doi.org/10.1007/978-0-387-72796-7_8
McDonald J.B., and J.W. Triplett 2024 “gintreg: A generalization of Statas intreg and stintreg Commands”, mimeo.
McDonald, J.B., and Y.J. Xu 1995 “A generalization of the beta distribution with applications”, Journal of Econometrics, 66(1-2): 133–152. [Erratum: Journal of Econometrics, 69, 427–428.]
McDonald, J.B., and Y. Xu 1992 “An empirical investigation of the likelihood ratio test when the boundary condition is violated”, Communications in Statistics: Simulation and Computation, 21(3): 879–892. doi.org/10.1080/03610919208813054
McDonald, J.B., J. Sorensen, and P.A. Turley 2011 “Skewness and Kurtosis Properties of Income Distribution Models”, Review of Income and Wealth, 59(2): 360–374. doi.org/10.1111/j.1475-4991.2011.00478.x
McManus, D.A. 1991 “Who Invented Local Power Analysis”, Econometric Theory, 7(2): 265–268. doi.org/10.1017/S026646660000445X
Nelson, F.D., and N.E. Savin 1988 “The Nonmonotonicity of the Power Function of the Wald Test in Nonlinear Models”, Department of Economics Working Paper Series No. 88–7, University of Iowa.
Nelson, F.D., and N.E. Savin 1990 “The Danger of Extrapolating Asymptotic Local Power”, Econometrica, 58(4): 977–981.
Rao, D.S.P., A.N. Rambaldi, G. Hajargasht, and W.E. Griffiths 2023 “The University of Queensland International Comparison Database, UQICD V3.0: User Guide”, CEPA Working Papers Series WP09/2022, School of Economics, University of Queensland, Australia.
Ruud, P.A. 2000 An Introduction to Classical Econometric Theory Oxford: Oxford University Press.
Sarabia, J.M., V. Jordá, and L. Remuzgo 2017 “The Theil Indices in Parametric Families of Income Distributions – A Short Review”, Review of Income and Wealth, 63(4): 867–880. doi.org/10.1111/roiw.12260
Savin, N.E., and A.M. Würtz 1999 “Power of Tests in Binary Response Models” Econometrica, 67: 413–421.
Singh, S.K., and G.S. Maddala 1976 “A function for the size distribution of incomes”, Econometrica, 44(2): 963–970.
StataCorp 2023 Stata Statistical Software: Release 18, College Station, TX: StataCorp LLC.
Vartia, P.L.I., and Y.O. Vartia 1980 “Description of the income distribution by the scaled F distribution model”, in N.A. Klevmarken and J.A. Lybeck (Eds.) The Statics and Dynamics of Income Clevedon, U.K. pp. 23–36.
Vogelsang, T.J. 1997 “Wald-type Tests for Detecting Shifts in the Trend Function of a Dynamic Time Series”, Econometric Theory, 13(6): 818–849. doi.org/10.1017/S0266466600006289
Vogelsang, T.J. 1999 “Sources of Nonmonotonic Power When Testing for a Shift in Mean of a Dynamic Time Series”, Journal of Econometrics, 88(2): 283–299. doi.org/10.1016/S0304-4076(98)00034-7
