The Singh–Maddala distribution: properties and estimation

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.

Appendix

By differentiating (5.2), the elements of the Fisher information matrix $I(\varTheta )$ for the parameters (α, β, λ) are:

$$I_{\alpha \alpha } = - \frac{n}{{\alpha^{2} }} - \alpha (\alpha + 1)(\lambda + 1)\sum\limits_{i = 1}^{n} {\frac{{(x_{i} /\beta )^{\alpha - 2} }}{{[1 + (x_{i} /\beta )^{\alpha } ]}}} - (\lambda + 1)\sum\limits_{i = 1}^{n} {\frac{{(x_{i} /\beta )^{2\alpha } \ln (x_{i} /\beta )}}{{[1 + (x_{i} /\beta )^{\alpha } ]^{2} }}} .$$

$$I_{\alpha \beta } = - \frac{n}{\beta } - \alpha (\lambda + 1)\sum\limits_{i = 1}^{n} {\frac{{(x_{i} /\beta )^{\alpha - 1} \ln (x_{i} /\beta )}}{{[1 + (x_{i} /\beta )^{\alpha } ]}}} + \alpha (\lambda + 1)\sum\limits_{i = 1}^{n} {\frac{{(x_{i} /\beta )^{2\alpha - 1} \ln (x_{i} /\beta )}}{{[1 + (x_{i} /\beta )^{\alpha } ]^{2} }}} .$$

$$I_{\alpha \lambda } = - \sum\limits_{i = 1}^{n} {\frac{{(x_{i} /\beta )^{\alpha } \ln (x_{i} /\beta )}}{{[1 + (x_{i} /\beta )^{\alpha } ]}}} .$$

$$\begin{aligned} I_{\beta \beta } & = \frac{n\alpha }{{\beta^{2} }} - \alpha (\lambda + 1)(\alpha + 1)\sum\limits_{i = 1}^{n} {\frac{{x_{i}^{\alpha } }}{{\beta^{\alpha + 2} [1 + (x_{i} /\beta )^{\alpha } ]}}} \\ & \quad + \alpha^{2} (\lambda + 1)\sum\limits_{i = 1}^{n} {\frac{{x_{i}^{2\alpha } }}{{\beta^{\alpha + 1} [1 + (x_{i} /\beta )^{\alpha } ]^{2} }}} . \\ \end{aligned}$$

$$I_{\beta \lambda } = - \alpha \sum\limits_{i = 1}^{n} {\frac{{x_{i}^{\alpha } }}{{\beta^{\alpha + 1} [1 + (x_{i} /\beta )^{\alpha } ]}}} .$$

$$I_{\lambda \lambda } = - \frac{n}{{\lambda^{2} }}.$$