Semi-empiricial Likelihood Confidence Intervals For The Differences Of Two Populations Based On Fractional Imputation

Empirical likelihood method proposed by Owen (Empirical likelihood ratio confidenceintervals for a single functional[J], Biometrika, 1988, 75: 237-249; Empirical likelihood con-fidence regions[J], The Annals of Statistics, 1990, 18: 90-120) to construct confidence inter-vals is an important statistical method. Qin Yongsong & Zhao Lincheng (Semi-parametriclikelihood confidence intervals for various differences of two populations[J], Statistics andProbability Letters, 1997, 33(2): 135-143; Empirical likelihood confidence intervals forquantile difference of two populations[J], Chinese Annals of Mathematics (Series A), 1997,18(6): 687-694; Semi-empirical likelihood confidence intervals for quantile differences ofsamples[J], Acta Mathematicae Applicatae Sinica, 1988, 21(1): 103-112; Empirical likeli-hood ratio confidence intervals for various differences of two populations[J], System Scienceand Mathematical Sciences, 2000, 13: 23-30) systematically study the construction of em-pirical likelihood confidence intervals for various differences of two populations under com-plete data. Missing data is common in daily life and scientific field, for example, in opinionpolls, market research surveys, medical studies and so on. The early way to handle missingdata is'Complete- Case'method, which deletes all missing data, and uses statistical methodsto the remaining data. A lot of useful information is lost in this way so that it could resultin wrong conclusion. A common method for handling incomplete data is to impute a valuefor each missing variables and then apply usual statistical methods to the'complete data'asif they are true observations. There are many methods to impute the missing data. Deter-ministic imputation and random imputation are common methods. Deterministic imputationcontains mean imputation, linear regression imputation and so on. Systematical discussionof imputation methods for missing data can be found in Little & Rubin (Statistical Analysiswith Missing Data[M], New York, John Wiley & Sons, 2002).Wang & Rao (Empirical likelihood for linear regression models under imputation formissing response[J], The Candian Journal of Statistics, 2001, 29: 597-608) obtain empiricallikelihood confidence intervals for the regression coefficient in a linear model with missing data. They use regression imputation method to fill missing data. Recently, Kim & Fuller(Fractional hot deck imputation[J], Biometrika, 2004, 91: 559-578) put forward a new im-putation method—-fraction imputation. Qin, Rao & Ren (Confidence intervals for marginalparameters under fractional linear regression imputation for missing data[J], Journal of Mul-tivariate Analysis, 2008, 99: 1232-1259) use the fraction imputation method to constructempirical likelihood confidence intervals for the response means, quantiles and distributionfunctions in a single linear model with missing data. In chapter 2 of this paper, we usefraction imputation to impute missing data and obtain'complete data'to construct semi-empirical likelihood confidence intervals for the differences of two population under MCARmechanism. In this way, we can improve the accuracy of the confidence intervals.Qin & Zhang (Empirical likelihood confidence intervals for differences between twodatasets with missing data[J], Pattern Recognition Letters, 2008, 29(6): 803-812) constructempirical likelihood confidence intervals for differences of two nonparametric populationswith MCAR mechanism. They use random imputation method to impute missing data. Inchapter 3, we use fraction imputation to impute data and construct semi-empirical likelihoodconfidence intervals for the differences of two population with mixed missing mechanism.We also can improve the accuracy of the confidence intervals in this situation. We found thatsemi-empirical likelihood confidence intervals for the differences of two populations basedon fraction imputation are not involved in existing literatures. This is the originality in thispaper.We summary some new findings in this paper as follows:1. Fractional imputation is used to fill in missing data under MCAR mechanism. Theasymptotic distributions of the semi-empirical likelihood ration statistic are obtained usingthe'complete data'after imputation. Especially, it is a random imputation method if the re-peated time is 1. Fractional imputation could reduce imputed variance and improve accuracyof statistic inference as the repeated time increases.2. We use fraction imputation to impute data and construct semi-empirical likelihoodconfidence intervals for the differences of two population with mixed missing mechanism.