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Confidence Intervals for the Population Mean Tailored to Small Sample Sizes, with Applications to Survey Sampling

• Michael Rosenblum, Center for AIDS Prevention Studies, Department of Medicine, University of California, San Francisco
• Mark J. van der Laan, University of California - Berkeley

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Abstract:

The validity of standard confidence intervals is based on the central limit theorem. For small sample sizes, the central limit theorem may give a poor approximation, resulting in confidence intervals that are misleading. We discuss this issue and propose methods for constructing confidence intervals for the population mean tailored to small sample sizes.

Our main contribution is presenting a simple approach for constructing confidence intervals for the population mean based on tail bounds for the sample mean that are correct for all sample sizes. Bernstein's inequality provides one such tail bound. The resulting confidence intervals have guaranteed coverage probability under much weaker assumptions than are required for standard methods. One of our methods not only has such guaranteed coverage probability for all sample sizes, but also produces confidence intervals whose widths are approximately equal to the widths of normal-based confidence intervals for large sample sizes. We show how to extend our approach to more general estimation problems as well. We describe how these methods can be used to obtain more reliable confidence intervals in survey sampling. As a concrete example, we construct a Bernstein-based confidence interval for the number of violent deaths between March 2003 and July 2006 in Iraq, based on data from the study ''Mortality after the 2003 invasion of Iraq: A cross-sectional cluster sample survey'', by Burnham et al. (2006).

Subject Area:

Design of Experiments and Sample Surveys, General Biostatistics, Statistical Models, Statistical Theory and Methods

Suggested Citation:

Michael Rosenblum and Mark J. van der Laan, "Confidence Intervals for the Population Mean Tailored to Small Sample Sizes, with Applications to Survey Sampling" (June 2008). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 237.
http://www.bepress.com/ucbbiostat/paper237