(untitled) (Communications in Mathematical Research, 2021)
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.