What can be said about the relationship between Mallow's Cp and Akaike information criterion?

Mallow's Cp and the Akaike Information Criterion (AIC) are both criteria used for model selection, and their relationship is rooted in the concepts of model fit and complexity. Both approaches aim to balance the goodness of fit of a model with a penalty for the number of parameters, thus helping to avoid overfitting.

Mallow's Cp is particularly designed for linear regression models and assesses how well a model with a certain number of predictors approximates the true model, comparing the residual errors of that model against a full or saturated model. It incorporates the idea of bias and variance in estimating the true error of a model.

In contrast, AIC is more general and applies to a wide variety of statistical models. Similar to Mallow's Cp, it considers the likelihood of the model and introduces a penalty term based on the number of parameters, making it a common tool in both nested and non-nested model comparisons.

The proportional similarity between Mallow's Cp and AIC arises because both criteria assess models using a balance of fit and complexity, despite their specific formulations and slight differences in how they penalize complexity. This relationship indicates that when one criterion improves, the other is likely influenced accordingly, capturing the overall essence of model selection in statistics.