## The Annals of Statistics

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

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

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Prediction region conditional coverage probability local asymptotic minimax convolution representation

#### Citation

Beran, Rudolf. Controlling Conditional Coverage Probability in Prediction. Ann. Statist. 20 (1992), no. 2, 1110--1119. doi:10.1214/aos/1176348673. https://projecteuclid.org/euclid.aos/1176348673