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The Singh–Maddala distribution: properties and estimation

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  • Devendra Kumar

    (Central University of Haryana)

Abstract

The Singh–Maddala distribution is very flexible and most widely used for modeling the income, wage, expenditure and wealth distribution of the country. Several mathematical and statistical properties of this distribution (such as quantiles, moments, moment generating function, hazard rate, mean residual lifetime, mean deviation about mean and median, Bonferroni and Lorenz curves and various entropies) are derived. We establish relations for the single and product moments of generalized order statistics from the Singh–Maddala distribution and then we use these results to compute the first four moments and variance of order statistics and record values for sample different sizes for various values of the shape and scale parameters. For this distribution, two characterizing results based on conditional moments of generalized order statistics and recurrence relations for single moments are established. The method of maximum likelihood is adopted for estimating the unknown parameters. For different parameters settings and sample sizes, the various simulation studies are performed and compared to the performance of the Singh–Maddala distribution. An application of the model to a real data set is presented and compared with the fit attained by some other well-known two and three parameters distributions.

Suggested Citation

  • Devendra Kumar, 2017. "The Singh–Maddala distribution: properties and estimation," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 8(2), pages 1297-1311, November.
    • Handle: RePEc:spr:ijsaem:v:8:y:2017:i:2:d:10.1007_s13198-017-0600-1
    DOI: 10.1007/s13198-017-0600-1
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    References listed on IDEAS

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    1. Singh, S K & Maddala, G S, 1976. "A Function for Size Distribution of Incomes," Econometrica, Econometric Society, vol. 44(5), pages 963-970, September.
    2. Mahesh Chandra & Nozer D. Singpurwalla, 1981. "Relationships Between Some Notions Which are Common to Reliability Theory and Economics," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 113-121, February.
    3. Sen, Amartya, 1973. "On Economic Inequality," OUP Catalogue, Oxford University Press, number 9780198281931.
    4. Kundu, Debasis & Howlader, Hatem, 2010. "Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data," Computational Statistics & Data Analysis, Elsevier, vol. 54(6), pages 1547-1558, June.
    5. Doiron, Denise J & Barrett, Garry F, 1996. "Inequality in Male and Female Earnings: The Role of Hours and Wages," The Review of Economics and Statistics, MIT Press, vol. 78(3), pages 410-420, August.
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    Cited by:

    1. Devendra Kumar & Anju Goyal, 2019. "Generalized Lindley Distribution Based on Order Statistics and Associated Inference with Application," Annals of Data Science, Springer, vol. 6(4), pages 707-736, December.

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