Empirical likelihood confidence bands for distribution functions with missing responses |
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Authors: | Qihua Wang Yongsong Qin |
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Affiliation: | 1. Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100080, China;2. School of Mathematical Sciences, Guangxi Normal University Guilin, Guangxi 541004, China |
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Abstract: | Distribution function estimation plays a significant role of foundation in statistics since the population distribution is always involved in statistical inference and is usually unknown. In this paper, we consider the estimation of the distribution function of a response variable Y with missing responses in the regression problems. It is proved that the augmented inverse probability weighted estimator converges weakly to a zero mean Gaussian process. A augmented inverse probability weighted empirical log-likelihood function is also defined. It is shown that the empirical log-likelihood converges weakly to the square of a Gaussian process with mean zero and variance one. We apply these results to the construction of Gaussian process approximation based confidence bands and empirical likelihood based confidence bands of the distribution function of Y. A simulation is conducted to evaluate the confidence bands. |
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Keywords: | Imputation Inverse probability weighting Confidence bands Empirical likelihood Missing at random |
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