Statistical Inference For Lorenz Curve With Nonignorable Missing Data

The Lorenz curve is a graphical representation of income inequality or wealth inequality,and it shows the percentage of total income earned by cumulative percentage of the population within a country or economic region.When conducting a survey of income,higher-income groups and lower-income groups are more likely to refuse to answer some income-related questions because of privacy concerns.This makes the income data subject to missingness,and the missing data mechanism is nonignorable,that is,the response probability model depends on the missing variables.This paper focuses on the statistical inference for Lorenz curve under the nonignorable missing mechanism.We assume that the missing mechanism is of the Logistic regression model,and employ the maximum likelihood method and the semi-parametric empirical likelihood method respectively to obtain a class of consistent estimators of the assumed parametric non-ignorable propensity score model.To estimate the Lorenz curve,the quantile of income data should be estimated in advance.Under the assumption that the missing-data mechanism is non-ignorable,consistent estimation of quantile function is challenging,and the results based on completecase analysis is invalid for this scenario.To this end,this paper proposes using inverse probability weighting and non-parametric kernel regression approaches for asymptomatically unbiased estimation of quantile function with non-ignorable missing data.Given the consistent estimators of non-ignorable propensity model and quantile function,a class of consistent estimators of Lorenz curve are constructed by applying inverse probability weighting approach.With simple random sampling,we conduct extensive simulation studies to assess the performance of all the proposed methods.The simulation results imply that all the proposed methods can produce robust and reliable results under all the considered scenarios,and the weighting approach based on the empirical likelihood method has the best performance among all the proposed methods.These results indicate that incorporating the auxiliary information into the inverse probability weighting procedure can improve the estimation efficiency for Lorenz curve with non-ignorable missing data.We also employ the method on a real data set as an illustration.