Influence function-based empirical likelihood and generalized confidence intervals for the Lorenz curve
- PDF / 1,086,272 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 35 Downloads / 221 Views
Influence function-based empirical likelihood and generalized confidence intervals for the Lorenz curve Yuyin Shi1 · Bing Liu1 · Gengsheng Qin1 Accepted: 23 June 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper aims to solve confidence interval estimation problems for the Lorenz curve. First, we propose new nonparametric confidence intervals using the influence functionbased empirical likelihood method. We show that the limiting distributions of the empirical log-likelihood ratio statistics for the Lorenz ordinates are standard chi-square distributions. We also develop “exact” parametric intervals for the Lorenz ordinate based on generalized pivotal quantities when the underlying income distribution is a Pareto distribution or a Lognormal distribution. Extensive simulation studies are conducted to evaluate the finite sample performances of the proposed methods. Finally, we apply our methods to a real income dataset. Keywords Empirical likelihood · Influence function · Generalized pivotal quantities · The Lorenz curve
1 Introduction Much attention has been given to rising income polarization and its varied implications in the US and across the world. For any economic policymaker, the estimation accuracy of the degree of income inequality is crucial for making informed decisions. One widely used tool to summarize the income distribution and, relatedly the degree of income inequality, is the Lorenz curve, which shows the percentage of the total income that the bottom (100 ∗ t)% (t ∈ [0, 1]) of households have. Figure 1 illustrates the Lorenz curve, in which the diagonal line at the 45◦ angle represents perfect income equality for all households. For a given t, the further away the Lorenz ordinate is from the diagonal, the more unequal the income distribution is. More formally, denote X as the random variable that is household income, and F(x) as its distribution function, i.e., F(x) represents the proportion of the population whose
B 1
Gengsheng Qin [email protected] Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, USA
123
Y. Shi et al. Fig. 1 Lorenz curve
income is less than or equal to x. Following Gastwirth (1971), a general definition of the Lorenz curve is below: assuming that F(x) is differentiable, 1 η(t) = μ
ξt
xd F(x), t ∈ [0, 1],
(1)
0
where μ is the mean of F, and ξt = F −1 (t) = in f {x : F(x) ≥ t} is the t-th quantile of F. For a fixed t ∈ [0, 1], the Lorenz ordinate η(t) is the ratio of the mean income of the lowest t-th fraction of households and the mean income of total households. Empirical analysis of economic inequality has commonly used the Lorenz curve. For instance, Atkinson (1970) provided a theorem relating the social welfare function to a partial ranking of income distributions according to the Lorenz curve criterion, and Doiron and Barrett (1996) used the Lorenz dominance as a fundamental concept for comparing income distributions. Besides applications in economics, the Lorenz curve is also widely used in other di
Data Loading...
