Adaptive testing for the partially linear single-index model with error-prone linear covariates

Adaptive testing for the partially linear single-index model (PLSIM) with error-prone linear covariates is considered. This is a fundamentally important and interesting problem for the current model because existing literature often assumes that the model structure is known before making inferences. In practice, this may result in an incorrect inference on the PLSIM. In this study, we explore whether the link function satisfies some special shape constraints by using an efficient penalized estimating method. For this we propose a model structure selection method by constructing a new testing statistic in the current setting with measurement error, which may enhance the flexibility and predictive power of this model under the case that one can correctly choose an adaptive shape and model structure. The finite sample performance of the proposed methodology is investigated by using some simulation studies and a real example from the Framingham Heart Study.