An analysis of the BOOTSTRAP method for estimating the mean squared error of statistical estimators

Abstract

One of the most crucial problems in theoretical and applied statistics is to determine the precision of the estimates produced by different statistical estimators. This problem is greatly increased when the population parametric characteristics are not known. Parallel to this problem is that of deciding how large (or small) the sample population must be in order to obtain a desire precision within certain range. There are several non-parametric methods to approach the first problem. The BOOTSTRAP Method (Efron, 1979) is one of these approaches and the one of interest in this thesis. With this method, one could improve the precision of the estimates and gain information about the distributional characteristics of statistical estimators. The bootstrap method has been amply compared with other methods; the results show that the bootstrap method often produces more precise estimates (i.e., with smaller mean squared error) than competitors such as the JACKNIFE, SECTIONING and CROSS-VALIDATION. However, the results that have been obtained are based on large sample sizes and large numbers of "bootstrap" replications. This thesis analyzes the behavior of the BOOTSTRAP method when the number of bootstrap replications is small. It tries to identify any tradeoffs between sample size and the number of bootstrap replications required to attain a desired precision in the estimates produced in several particular situations. One of the goals is to produce graphical displays that will indicate to the experimental statistician the price that must be paid in the precision of the estimates, obtained with the bootstrap method, when sample size is small, and the number of bootstrap replications to use in this situation.