This study proposes a reverse Pareto model to describe the power-law behavior for the lower tail data of income distribution and illustrates an application on Malaysian and Italian household income data. A robust method based on probability integral transform statistic is used for estimating the shape parameter of the reverse Pareto model to allow for the existence of outlying observations in the lower tail data. Besides that, the optimal threshold of reverse Pareto is determined by using Kolmogorov–Smirnov statistic. It is found that the fitted reverse Pareto adequately describes the lower tail data of both datasets, suggesting that the power-law behavior is obeyed. In addition, the estimated optimal threshold of reverse Pareto model can be utilized as an alternative measure for the relative poverty line. Based on the reverse Pareto model, the Lorenz curve, Gini and Theil coefficients are determined, and it is found that low income inequality is observed for the period of the study. The fitted Lorenz curve shows that nearly 80% of the total household income is owned by the bottom 80%, whereas the remaining total household income is owned by the top 20%. Finally, comparison of the reverse Pareto model with some alternative distributions such as shifted reverse exponential, shifted reverse stretched exponential and shifted reverse lognormal in terms of model fitting for the lower tail data is also conducted. The results show that the reverse Pareto model outperform all the other models.

这项研究提出了一个反向帕累托模型来描述收入分配的低尾数据的幂律行为，并说明了在马来西亚和意大利家庭收入数据中的应用。基于概率积分变换统计量的鲁棒方法用于估计反向帕累托模型的形状参数，以允许在下尾数据中存在离群观测值。除此之外，使用Kolmogorov–Smirnov统计量确定反向Pareto的最佳阈值。发现拟合的反向帕累托充分描述了两个数据集的下尾数据，表明服从了幂律行为。此外，反向帕累托模型的估计最佳阈值可以用作相对贫困线的替代度量。根据反向帕累托模型，确定了Lorenz曲线，Gini和Theil系数，发现在研究期间观察到低收入不平等。拟合的洛伦兹曲线显示，最低的80％拥有近80％的家庭总收入，而最高的20％拥有剩余的总家庭收入。最后，还对反向帕累托模型与一些替代分布进行了比较，例如在对下尾数据的模型拟合方面，偏移了反向指数，偏移了反向拉伸指数和偏移了对数正态。结果表明，反向帕累托模型优于所有其他模型。拟合的洛伦兹曲线显示，最低的80％拥有近80％的家庭总收入，而最高的20％拥有剩余的总家庭收入。最后，还对反向帕累托模型与一些替代分布进行了比较，例如在对下尾数据的模型拟合方面，偏移了反向指数，偏移了反向拉伸指数和偏移了对数正态。结果表明，反向帕累托模型优于所有其他模型。拟合的洛伦兹曲线显示，最低的80％拥有近80％的家庭总收入，而最高的20％拥有剩余的总家庭收入。最后，还进行了反向帕累托模型与一些替代分布的比较，例如在对下尾数据的模型拟合方面，偏移了反向指数，偏移了反向拉伸指数和偏移了对数正态。结果表明，反向帕累托模型优于所有其他模型。还针对下尾数据进行了模型拟合方面的移位反向拉伸指数和移位对数正态。结果表明，反向帕累托模型优于所有其他模型。还针对下尾数据进行了模型拟合方面的移位反向拉伸指数和移位对数正态。结果表明，反向帕累托模型优于所有其他模型。