  When performing supervised learning with the model selected using validation
error from sample splitting and cross validation, the minimum value of the
validation error can be biased downward. We propose two simple methods that use
the errors produced in the validating step to estimate the test error after
model selection, and we focus on the situations where we select the model by
minimizing the validation error and the randomized validation error. Our
methods do not require model refitting, and the additional computational cost
is negligible. In the setting of sample splitting, we show that, the proposed
test error estimates have biases of size $o(1/\sqrt{n})$ under suitable
assumptions. We also propose to use the bootstrap to construct confidence
intervals for the test error based on this result. We apply our proposed
methods to a number of simulations and examine their performance.
