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U-Statistics for some nonparametric incomplete data models with side information

Ao Yuan, Ming Xiong, Ming T. Tan


We study a class of U-statistics for incomplete data nonparametric models, with side information incorporated. We concentrate on five commonly used incomplete data models: convolution model, multiplicative censoring model, types I and II interval censoring models,
and the double censoring model. In these models, often the original data are unobserved or only partially observed. Our interest is to estimate, via U-statistic, the parameter of interest which is a functional of the observed data distribution, but the side information is given by a set of constraints on the original data distribution. Different from most existing methods for incorporating side information with a set of given constraints on the distribution of the fully observed data, since the original data is not fully observed, these constraints are not directly usable. To make use of these constraints on the observed data, we use a conditional constraints which is the conditional expectation of the original constraints given the observed data, thus the observed data are directly incorporated into the constraints, and with the available side information incorporated, the proposed U-statistic is more efficient than that without such information. The method can be applied to many more incomplete/missing data models. Some basic properties of the proposed statistics are investigated.


Conditional constraints, incomplete data model, NPMLE, side information, Ustatistics.

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