# A new path from p-value to LOOCV

I have been thinking about a path from p-value to Leave One Out Cross Validation (LOOCV). I figure if I can get to AIC then I am good as AIC is asymptotically equivalent to LOOCV.

This is the basic flow.

pvalue -> wilks statistic -> KL distance -> AIC -> LOOCV

1. p-value can be linked to the LRT from the Wilk’s Likelihood ratio statistic $$\text{P-Value} = P(W\geq-2log(R))$$ Where ‘R’ is the the likelihood ratio $\frac{L(\hat{\theta})}{L(\theta_0)}$.

2. Pawitan (2001), in his book, showed that minimizing the AIC corresponds to minimizing the KL distance. KL distance is

$$D(g,f) = E_{g}\text{log}\frac{g(X)}{f(X)}$$

For discrete distributions we have that $g(x_i) = P(X=x_i)$. This makes the KL distance simply

$$D(g,f) = \Sigma_{i}g(x_i)\text{log}\frac{g(x_i)}{f(x_i)}$$

If $g(x_i)$ is associated with observed frequencies, and $f(x_i)$ is associated with the model-based frequences we have the familiar goodness-of-fit statistic.

$$W=\Sigma O\text{log}\frac{O}{E}$$

Where W is the likelihood ratio statistic. I think this can then be linked back to point (1). Someone else will have to verify this and show how it can be generalized.