In classical model-based inferences, hypothesis testing and confidence intervals are proved to be valid under the assumption that model selection and data analysis are independent processes and can be treated separately. However, in practice, a two-step method is often involved and the model is selected by data-driven procedures. Therefore, we cannot simply assume the selected model is correct.

In recent decades, many researchers have been aware of the post-selection problem. Empirical results have shown that when the model selection results are very unstable, the resulting coverage probabilities can be far below the desirable level, regardless of the sample size.

I have developed our own R packages that are convenient for the general users to implement the relatively complex post-model-selection inference methods. Relative R packages can be downloaded from https://github.com/liruohan1436311/Mixed_Residual_Bootstrap