Empirical Likelihood Inference For Quantile Differences Of Two Populations With Missing Data

Empirical likelihood method proposed by Owen (1988, 1900) is one of the mostimportant tools in statistical field. It is a hot topic to use empirical likelihood toconstruct confidence intervals for unknown parametric in the study field of interval es-timation under complete sample. However due to all kinds of reasons, we often obtainsample with missing data in reality. In this situation, the usual empirical likelihoodmethod cannot be applied directly. A common method to handle sample with miss-ing data is to impute a value for each missing data, to obtain'complete'sample ofpopulations. After that, we use usual empirical likelihood method to make inferencebased on'complete'sample. Fractional imputation proposed by Kim & Fuller (2004)have some nice features in the current popular imputation methods. For example,the method can decrease the imputation variance. Fractional imputation method isthus studied widely by many authors. Recently Qin, Rao & Ren (2008) have studiedempirical likelihood confidence intervals for the quantile of response variable in a lin-ear model based on fractional imputation. On the other hand, Qin Yongsong (1997)studied empirical likelihood confidence intervals for quantile di?erences of two popu-lations under complete sample. However all papers published didn't study quantiledi?erences of two populations with missing data. Therefore we will discuss this topicin this paper. Empirical likelihood confidence intervals are obtain for quantile di?er-ences of two nonparametric populations with missing data and quantile di?erences forresponse variables in two linear regression models with missing data,the main resultsare summarized as follows:1. We introduce empirical likelihood method into empirical likelihood confidenceintervals for the quantile di?erences of two nonparametric populations with missingdata, where two populations are missing completely at random (MCAR). First, frac-tional imputation method is used to impute the missing data, to obtain'complete'data. Then we can prove the asymptotic distributions of the empirical likelihood ra-tios statistics for quantile di?erences based on the'complete'sample are scalarχ12. The results can be used to construct empirical likelihood confidence intervals for thequantile di?erences ?.2. We introduce empirical likelihood method into two liner models with missingdata, and study empirical likelihood confidence intervals for the quantile di?erences ofresponse variables, where response variables are missing at random (MAR). We usefractional linear regression imputation method to impute the missing data of responsevariables, and obtain'complete'data of two linear regression models. Under someconditions, it is proved that the asymptotic distributions for the empirical likelihoodratios of quantile di?erences ? of response variables are scaledχ12. Empirical likeli-hood confidence intervals for q-th quantile di?erences ? of response variables are thenconstructed based on these results.We summary some new findings in this paper as follows:Firstly, under incomplete data and MCAR missing mechanism, we use fractionalimputation method (a repeated imputation method) to impute missing data,andobtain empirical likelihood confidence intervals for quantiles di?erences of two non-parametric populations. The usual (single) imputation method is a special case offractional imputation. As the repeated time increases, fractional imputation can reducethe imputation variance. Comparing with single imputation, fractional imputation canimprove the accuracy of the confidence intervals.Secondly, under incomplete data and MAR missing mechanism, we use fractionalimputation method to impute missing data,and obtain empirical likelihood confidenceintervals for quantiles di?erences of response variables in two liner models. MAR is aweaker restriction than MCAR, and MAR is easy to be satisfied in real applications.