Semi-Empiricial Likelihood Confidence Intervals for the Differences of Two Populations Based on Fractional Imputation

Abstract

Suppose that there are two populations $x$ and $y$ with missing data on both of them, where $x$ has a distribution function $F(·)$ which is unknown and $y$ has a distribution function $G_θ(·)$ with a probability density function $g_θ(·)$ with known form depending on some unknown parameter $θ$. Fractional imputation is used to fill in missing data. The asymptotic distributions of the semi-empirical likelihood ration statistic are obtained under some mild conditions. Then, empirical likelihood confidence intervals on the differences of $x$ and $y$ are constructed.