Semi-empirical Likelihood Inference For Quantile Diffrences Of Two Populations With Missing Data By Using Fractional Imputation

Item non-response occurs frequently in daily life. It happens in many fields, suchas opinion polls, medical studies and market research surveys for various reasons, andthe usual inferential procedures for complete data sets cannot be applied directly insuch circumstances. It needs to do some treatment on the data sets before we apply theusual inferential procedures. Statistical inference for missing data is an important andpopular research field (Little and Rubin, Statistical analysis with missing data[M], NewYork: John Wiley and Sons, 2002.). The Complete-Case method was presented firstlyto treat the incomplete data sets. It delete all of non-response items at first, and thenconsider the rest as a'complete'data set to which the usual inferential procedures willbe applied. But the number of the sample will be cut down and the deviation of the datawill lead to a wrong conclusion easily. A common method to treat the incomplete datasets is to impute values for each missing response so that we can obtain a'complete'data set and apply the usual inferential procedures, and this sort of method is dividedinto two– deterministic imputation method and stochastic imputation method. Basingon the'complete'data set, the former cannot make statistic an approaching one exceptaverage statistic, but the latter will do.Studies of comparison of various di?erences of populations is an important topic inmany fields, such as in market research surveys, economical and educational fields. QinYongsong & Zhao Lincheng (Semi-parametric likelihood confidence intervals for vari-ous di?erences of two populations[J]. Statistics and Probability Letters, 1997, 33(2):135-143; Empirical likelihood confidence intervals for quantile di?erences of two popu-lations [J]. Chinese Ann Math(Ser A), 1997, 18(6): 687-694; Semi-empirical likelihoodconfidence intervals for quantile di?erences of two samples [J]. Acta Mathematicae Ap-plicatae Sinica, 1998, 21(1): 103-112; Empirical likelihood ratio confidence intervals forvarious di?erences of two populations[J]. System Science and Mathematical Sciences,2000, 13: 23-30.) systematically study the construction of EL confidence intervals for various di?erences of two populations under complete data. By using the single stochas-tic imputation (a special case of fractional imputation) to impute values for each missingresponse, Zhang Junchao (Empirical Likelihood Inference for The Di?erence Indicatorof Two Samples With Missing Data[D]. Guilin, Guangxi Normal University, 2007.)construct EL confidence interval for quantile di?erences in semi-parameter model withmissing data under the MCAR mechanism. But the imputation variance is somewhatbigger than what we prefer to. In order to reduce the imputation variance, in chapter2 of this paper, under MCAR missing mechanism, we obtain'complete'data set byusing fractional imputation method, and prove that the approaching distribution of thesemi-empirical likelihood ratio statistic for quantile di?erences is a weighted chisquaredistribution. Accordingly, we construct a semi-empirical likelihood confidence intervalfor quantile di?erences between a nonparametric population and a parametric popula-tion, which heavily improve the cover precision of the confidence intervals. Consideringthat the restriction of the mechanism of missing at random (MAR) is weaker than thatof MCAR and MAR is easy to be satisfied in real applications, in chapter 3 of thispaper, we generalize the main result of chapter 2 to the case of mixing use of missingmechanism. We obtain'complete'data set by using fractional (regression) imputation,and construct a semi-empirical likelihood confidence intervals for quantile di?erencesbetween a nonparametric population under MAR missing mechanism and a parametricpopulation under MCAR missing mechanism .Here we summary some new findings in this paper.1. Under incomplete data and MCAR missing mechanism, we use fractional impu-tation method (a repeated imputation method) to impute missing data,and constructa semi-empirical likelihood confidence intervals for quantile di?erences between a non-parametric population and a parametric population. The single stochastic imputationmethod is a special case of fractional imputation. As the repeated time of imputationincreases, fractional imputation can decrease the imputation variance. Comparing withsingle stochastic imputation, fractional imputation can improve the cover precision ofthe confidence intervals.2. Under incomplete data and mixing use of missing mechanism, we use fractionalimputation method to impute missing data,and construct a semi-empirical likelihoodconfidence intervals for quantile di?erences between a nonparametric population underMAR missing mechanism and a parametric population under MCAR missing mecha-nism. MAR is a weaker restriction than MCAR, and MAR is easy to be satisfied inreal applications.