Statistical Inference For Estimating Equations Statistical Models With Missing Data

The problem of missing data occurs commonly in daily life, It happens in market re-search,census,reliability life test and the flight recorder system and so on,it brings a lot of diffi-culties to data analysis and application. Large amounts of missing data may be generated for various reasons such as non-controlled or human error causes.Many classical statistical models and statistical methods are applied to complete data, statistical research is relatively small in the case of missing data.thus,statisticians and practical worker have great interest on statistical in-ference for missing data which is an important research field(Little and Rubin(2002). In such circumstances of missing data, it needs to do some treatments on data before we can use usual statistical approaches,a common method is to impute values base on the original data in order to obtain a "complete data" set, then apply the complete datal of statistical methods.This paper mainly studies the EL inference for parameters defined by estimating equations in the presence of missing data, the EL method of Qin and Lawless(1994) is more mature approach on the estimating equation under complete data.Firstly,they constructs EL ratio function of the param-eters,then they proved that the limiting distribution of EL ratio function is chi-square distribution under a null hypothesis. Nonparametric imputation method of Wang and Chen(2009) is a classic approach on the estimating equation under MAR missing mechanism.But with their method, the limiting distribution of test statistic is a weighted chi-squared distribution with unknown weights under the null hypothesis,they needs to estimate the adjusting coefficient in the actual applica-tion.For their insufficient section,we propose to construct EL confidence intervals for the paramet-ric of interest defined by the estimating equations,this method is based on "complete data" after inverse probability weighted imputation,we can obain the MLE ofθbased on the EL ratio func-tion and prove the asymptotic normality under certain conditions,then we can obain the conclusion that empirical likelihood function of the test statistic at the time of the test H0:θ=θ0(θ0 is com-pletely known) is asymptotically chi-square distribution with r degrees of freedom under the null hypothesis (where r is the estimated number of equations).These results can be used to obtain EL confidence intervals(regions) for the parameterθ. We also do a simulation study,the result shows that constructing EL confidence intervals using inverse probability weighted imputation has good coverage probability under small samples.According to above-mentioned,the coverage probability of our method is better than Wang and Chen's under the same levels and capacity, the interval length is gradually reduced as the sample size increased based on our method. Here we summary some new findings in our work.1.Firstly,this paper weakens the conditions in the Wang and Chen (2009)(this mainly embod-ies in enlarging applicable range of kernel estimator.),expands the applicable scope of the model.2.1n studying the construction of confidence intervals(regions) for the parametric of inter-est defined by the estimating equations with random missing data,we use the inverse probability weighted imputation method,based on this imputation method, constructing EL ratio function which is asymptotically chi-squared distribution.These results are used to obtain EL confidence intervals(regions) for the parametric of interest defined by the general estimating equations,and no adjustment is needed at the time of constructing confidence intervals. This would improve the accuracy of the EL confidence intervals.