5.3 Leave-One-Out Cross-Validation (LOOCV)

• LOOCV aims to address some of the drawbacks of the validation set approach.

• Similar to validation set approach, LOOCV involves splitting the data into a training set and validation set.

• However, the validation set includes one observation, and the training set includes $n-1$ observations. This process is repeated for all observations such that $n$ models are estimated.

  • Having a large training set avoids the problems from not using all (or almost all) of the data in estimating the model.
  • Conversely, the validation error for a given model is highly variable since it consists of one observation, although it is unbiased.

LOOCV estimate of test error is averaged over the $n$ models:

$$CV_{n} = \frac{1}{n}{\sum_{i=1}^{n}}(y_{i}-\hat{y_{i}})^2$$