## The Annals of Statistics

### Controlling Conditional Coverage Probability in Prediction

Rudolf Beran

#### 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 $n^{-1/2}$.

#### Article information

Source
Ann. Statist., Volume 20, Number 2 (1992), 1110-1119.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348673

Digital Object Identifier
doi:10.1214/aos/1176348673

Mathematical Reviews number (MathSciNet)
MR1165609

Zentralblatt MATH identifier
0746.62095

JSTOR