Empirical Likelihood Confidence Intervals For The Differences Of Quantiles With Missing Data

For making statistical inference it is typically assumed that all the responses in the sample are available.This may not hold true in many practical situations and some responses may be missing for various reasons such as unwillingness of some sampled units to supply the desired information ,loss of information caused by uncontrollable factors, failure on the part of the inves-tigator to gather correct information, or natural disasters, plane crash, etc.In fact, the missing data Phenomenon in polls, market research, E-mail inquiries, medical research and other scientific experiments often Appear. In this case, usually the reasoning methods cannot directly applica-ble. The existing methods usually use Empirical likelihood will non-parameter sample parametric method to solve the problem.When the overall or model of information missing data, make its Become the complete data, using standard statistical methods for "complete data" statistical inference.Commonly used imputation methods include deterministic imputation and random imputa-tion.We do not use the deterministic imputation as it does not proper in making inference for distribution functions.Suppose that there is a nonparametric populations x with missing data on it. We are inter-ested in constructing confidence intervals on the quantile differences of x. Random imputation is used to fill in the missing data. Empirical likelihood confidence intervals on the differences are constructed.The original idea of empirical likelihood dates back to Hartley and Rao in sample survey con-text, and to Thomas and Grunkemeier in survival analysis context. Owen made a systematic study of the empirical likelihood method in the complete data settings. It has several advantages over the normal-approximation based methods and the bootstrap in constructing confidence intervals.Wang and Rao first use EL method to construct confidence intervals for the mean of the response variable in a linear model with missing data. Wang and Rao then use EL method to construct confidence intervals for the mean of the response variable in a nonparametric regression model with missing dataIn this study a practical problem rich-poor gap, we need to understand the rich society income And poor income differences. The income difference value cannot too big, otherwise they will be on the economy and social harmony Adverse effects, this is our research points digits. An applica-tion of the difference Points to understand sample median differences structure is very Important, has been used widely in social, economic, medical each kind of domain.The work of this paper and innovation:We introduce empirical likelihood method into empirical likelihood confidence intervals for the quantile differences of single population with missing data, where population are missing com-pletely at random(MAR). First, random impulation method is used to impute the missing data, to obtain'complete'data.Then we can prove the asymptotic distributions of the empirical likelihood ratios statistics for quantile differences based on the'complete'sample are scalarχ12.The results can be used to construct empirical likelihood confidence intervals for the quantile differences.For the quantile differences, Qin Yongsong(1997) studied empirical likelihood confidence in-tervals for quantile differences of two populations under complete sample,and Wang Lirong(2008) studied empirical likelihood inference for quantile differences of two populations under with miss-ing data.Because the model is studied two overall, there are two distribution, might as well the distribution of one of the general assumption for G, another General distribution for F, the above two scholars who studied the same two overall their distribution of distribution probabil-ity of differences. Portion digits Empirical likelihood statistical inference (such as Delta= G-1(q) - F-1(q)empirical likelihood statistical inference). And his research is single Overall, namely the same distribution of different distribution probability of the points digits difference (such as Delta= F-1(q) - F-1(p)empirical likelihood Statistical inference), this is this article innovative points.