# 9. The Sufficient Statistic in Sampling

Published Online: 10 FEB 2012

DOI: 10.1002/9781118162934.ch9

Copyright © 2012 John Wiley & Sons, Inc.

Book Title

## Sampling, Third Edition

Additional Information

#### How to Cite

Thompson, S. K. (2012) The Sufficient Statistic in Sampling, in Sampling, Third Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9781118162934.ch9

#### Publication History

- Published Online: 10 FEB 2012
- Published Print: 23 FEB 2012

#### Book Series:

#### Book Series Editors:

- Walter A. Shewhart and
- Samuel S. Wilks

#### ISBN Information

Print ISBN: 9780470402313

Online ISBN: 9781118162934

- Summary
- Chapter

### Keywords:

- Hansen-Hurwitz estimator;
- Horvitz-Thompson estimator;
- probability-proportional-to-size sampling;
- random sampling;
- unequal probability sampling

### Summary

The minimal sufficient statistic for the finite-population survey sampling situation is the unordered set of distinct, labeled observations. Thus, the set of y-values alone is not enough—the identities of the associated units may be helpful in estimation. In unequal probability sampling without replacement, unconditional inclusion probabilities may be extremely difficult to compute. In principle, however, one need consider only estimators that are functions of the minimal sufficient statistic. For any estimator that is not a function of the minimal sufficient statistic, one may obtain (using the Rao–Blackwell method) an estimator, depending on the minimal sufficient statistic, that is as good or better. This chapter illustrates the results of minimal sufficiency in the sampling situation.

#### Controlled Vocabulary Terms

Hansen-Hurwitz estimator; Horvitz-Thompson estimator; probability sampling; random sampling