Conditional confidence intervals for the true classification error

In this paper, we consider the joint distribution of the true error and the estimated error, assuming a random feature-label distribution. From it, we derive the conditional expectation of the true error and the 95% upper confidence bound for the true error given the estimated error. Numerous classification and estimation rules are considered across a number of models. Although specific results depend on the classification rule, error-estimation rule, and model, some general trends are seen: (1) the conditional expected true error is larger (smaller) than the estimated error for small (large) estimated errors; and (2) the confidence bounds tend to be well above the estimated error for low error estimates, becoming much less so for large estimates.