Generalised Skew Log Laplace Distribution

Pareto distribution was formulated to characterise incomes. Generally it is observed that reported incomes are understatements of the true values. Hartley and Revankar (J Econom 2:327-341, 1974) developed a model for underreported income, which was named as Skew log Laplace distribution (SLLD) by Kozubowski and Podgorski (Int Math J 3:467-495, 2003). The main aim of this paper is to develop a distribution of reported income, when reported incomes are mixture of true and underreported incomes. We named this distribution as generalised skew log Laplace distribution (GSLLD) as SLLD is a particular case of GSLLD. The maximum likelihood estimators of the parameters (MLE) of GSLLD when scale parameter is known are obtained and it is proved that MLEs are best asymptotically normal. The confidence intervals of the parameters are provided. The underreported incomes can be analysed using GSLLD or SLLD. The likelihood ratio test is derived to choose between GSLLD and SLLD. Finally, an application of GSLLD to real life income values is presented.

Automated summary

The paper discusses the Skew log Laplace distribution and the generalised skew log Laplace distribution for analysing reported incomes that are mixtures of true and underreported incomes. It presents the maximum likelihood estimators, confidence intervals, and a likelihood ratio test for choosing between the distributions. An application of the distributions to real life income values is also provided.