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Consistent inference for biased sub-model of high-dimensional partially linear model. (English) Zbl 1209.62071
Summary: We study a working sub-model of partially linear models determined by variable selection. Such a sub-model is more feasible and practical in applications, but usually biased. As a result, the common parameter estimators are inconsistent and the corresponding confidence regions are invalid. To deal with the problems relating to the model bias, a nonparametric adjustment procedure is provided to construct a partially unbiased sub-model. It is proved that both the adjusted restricted-model estimator and the adjusted preliminary test estimator are partially consistent, which means when the samples drop into some given subspaces, the estimators are consistent. Luckily, such subspaces are large enough in a certain sense and thus such a partial consistency is close to global consistency. Furthermore, we build a valid confidence region for parameters in the sub-model by the corresponding empirical likelihood.
62G08Nonparametric regression
62G15Nonparametric tolerance and confidence regions
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