Parameter Estimation of Singh Maddala Distribution by Moments
Abstract
Singh Maddala Distribution is a flexible distribution and mostly used for modeling the income, wage, expenditure and for wealth distribution of a country. To model the density of the income distribution, we have derived L-moments and TL-moments of the SMD in closed form. These moments are used to estimate the scale parameter which is related to the inequality of the income distribution. This study shows that L-moments and TL-moments are better alternatives against conventional moments and estimators of these moments equally applicable for both small and large sample size. Through Monte Carlo simulation study, we find TL-moments estimators provide least bias and smaller Root Mean Square Error of statistic, coefficient of variation, skewness and kurtosis. We also find that TL-moments are superior to L-moments if we are allowed to discord extreme value, otherwise L-moments provide the best fit.About this article
References
Abdul-Moniem, I., and Youssef M. Selim, "TL-moments and L-moments estimation for the generalized Pareto distribution" Applied Mathematical Sciences. Vol.3, No.1 (2009), pp. 43-52.
Abdul-Moniem, Ibrahim B. "L-moments and TL-moments estimation for the exponential distribution." Far East Journal of Theoretical Statistics, Vol.23, No.1, (2007), pp.51-61.
Ahmad, Ummi Nadiah, Ani Shabri, and Zahrahtul Amani Zakaria, "TL-moments and L-moments Estimation of the Generalized Logistic Distribution", Journal of Mathematics Research,Vol. 3, No.1, (2011), pp.97-106.
Asquith, William H., "L-moments and TL-moments of the generalized lambda distribution", Computational Statistics & Data Analysis, Vol.51, No.9, (2007), pp. 4484-4496.
Atoda, Naosumi, Surura, T. and Tachibanaki, T., “Statistical Inference of Functional Forms for Income Distribution”. manuscript, (1980), Kyoto University.
View more references (13)
Dhongde, Shatakshee, and Camelia Minoiu, "Global poverty estimates: Present and future", Society for the Study of Economic Inequality, Working Paper No 181 (2010).
Elamir, Elsayed AH, and Allan H. Seheult, "Trimmed L-moments", Computational Statistics & Data Analysis, Vol. 43, No.3, (2003), pp. 299-314.
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, Vol.15, No.5, (1979), pp.1049-1054.
Hosking, J. R. M., and J. R. Wallis, "A comparison of unbiased and plotting-position estimators of L moments", Water Resources Research, Vol. 31, No.8, (1995), pp.2019-2025.
Hosking, Jonathan RM., "L-moments: analysis and estimation of distributions using linear combinations of order statistics", Journal of the Royal Statistical Society. Series B (Methodological), (1990), pp.105-124.
Kleiber, Christian, "Dagum vs. Singh-Maddala income distributions", Economics Letters, Vol. 53, No.3, (1996), pp.265-268.
Kundu, Debasis, and Mohammad Z. Raqab, "Generalized Rayleigh distribution: different methods of estimations", Computational statistics & data analysis, Vol. 49, No.1, (2005), pp.187-200.
McDonald, James B., "Some generalized functions for the size distribution of income", Econometrica: Journal of the Econometric Society, (1984), pp.647-663.
Shahzad, M. N, and Asghar, Z., "Comparing TL-Moments, L-Moments and Conventional Moments of Dagum Distribution by Simulated data", Revista Colombiana de Estadística, Vol. 36, No.1, (2013), pp.79-93.
Singh, S.K., and G.S. Maddala, “A Function for the size distribution of income”, Econometrica,Vol. 44, (1976), pp.963-970.
Vogel, Richard M., and Neil M. Fennessey, "L moment diagrams should replace product moment diagrams", Water Resources Research, Vol. 29, No.6, (1993), pp.1745-1752.
Ye, Yuan, Broderick O. Oluyede, and Mavis Pararai, "Weighted generalized beta distribution of the second kind and related distributions", Journal of Statistical and Econometric Methods, Vol.1, No.1, (2012), pp.13-31.
Zhu, Feng."A nonparametric analysis of the shape dynamics of the US personal income distribution”, BIS Working Papers, 184, (2005), pp.1962-2000".