Asymptotic normality and the bootstrap in stratified sampling. (English) Zbl 0542.62009

Bickel, P. J.; Freedman, D. A.

Ann. Stat. 12, 470-482 (1984).

The asymptotic distributions of the estimates of a linear combination of stratum means in stratified sampling are considered. Two situations are discussed: (a) The populations are assumed arbitrary and the sampling is with replacement. (b) The populations are assumed finite and the sampling is 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-statistics. Via a bound on the Mallows distance, situations will be identified in which the bootstrap approximation works even though the normal approximation fails.

Reviewer: Z.Lin

MSC:

62E20	Asymptotic distribution theory in statistics
62D05	Sampling theory, sample surveys
60F05	Central limit and other weak theorems

Keywords:

linear combination of stratum means; stratified sampling; with replacement; without replacement; Lindeberg conditions; asymptotic normality; consistency; variance estimators; bootstrap approximation; t- statistics; Mallows distance