Lilliefors, Kolmogorov-Smirnov and cross-validation

In statistics, Kolmogorov–Smirnov test is a popular procedure to test, from a sample [latex]\{x_1,\cdots,x_n\}[/latex] is drawn from a distribution [latex]F[/latex], or usually [latex]F_{\theta_0}[/latex], where [latex]F_{\theta}[/latex] is some parametric distribution. For instance, we can test [latex]H_0:X_i\sim\mathcal{N(0,1)}[/latex] (where [latex]\theta_0=(\mu_0,\sigma_0^2)=(0,1)[/latex]) using that test. More specifically, I wanted to discuss today [latex]p[/latex]-values. Given [latex]..