Controlling Conditional Coverage Probability in Prediction

Abstract

Suppose the variable $X$ to be predicted and the learning sample $Y_n$ that was observed are independent, with a joint distribution that depends on an unknown parameter $\theta$. A prediction region $D_n$ for $X$ is a random set, depending on $Y_n$, that contains $X$ with prescribed probability $\alpha$. In sufficiently regular models, $D_n$ can be constructed so that overall coverage probability converges to $\alpha$ at rate $n^{-r}$, where $r$ is any positive integer. This paper shows that the conditional coverage probability of $D_n$, given $Y_n$, converges in probability to $\alpha$ at a rate which usually cannot exceed $n^{-1/2}$.