## The Annals of Statistics

### Asymptotic Normality and the Bootstrap in Stratified Sampling

#### Abstract

This paper is about the asymptotic distribution of linear combinations of stratum means in stratified sampling, with and without replacement. Both the number of strata and their size is arbitrary. Lindeberg conditions are shown to guarantee asymptotic normality and consistency of variance estimators. The same conditions also guarantee the validity of the bootstrap approximation for the distribution of the $t$-statistic. Via a bound on the Mallows distance, situations will be identified in which the bootstrap approximation works even though the normal approximation fails. Without proper scaling, the naive bootstrap fails.

#### Article information

Source
Ann. Statist., Volume 12, Number 2 (1984), 470-482.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346500

Digital Object Identifier
doi:10.1214/aos/1176346500

Mathematical Reviews number (MathSciNet)
MR740906

Zentralblatt MATH identifier
0542.62009

JSTOR