Bootstrapping

Basic Principles and Estimation Procedures

The fundamental principle on which the procedure is based is the belief that under certain general conditions, the relationship between a bootstrapped estimator and a parameter estimate should be similar to the relationship between the parameter estimate and the unknown population parameter of interest. As a means of better understanding the origins of this belief, Peter Hall suggested a valuable visual: a nested Russian doll. According to Hall's thought experiment, a researcher is interested in [Page 102]determining the number of freckles present on the outermost doll. However, the researcher is not able to directly observe the outermost doll and instead can only directly observe the inner dolls, all of which resemble the outer doll, but because of their successively smaller size, each possesses successively fewer freckles. The question facing the researcher then is how to best use information from the observable inner dolls to draw conclusions about the likely number of freckles present on the outermost doll. To see how this works, assume for simplicity that the Russian doll set consists of three parts, the outermost doll and two inner dolls. In this case, the outermost doll can be thought of as the population, which is assumed to possess n0 freckles; the second doll can be thought of as the original sample, which is assumed to possess n1 freckles; and the third doll can be thought of as the bootstrap sample, which is assumed to possess n2 freckles. A first guess in this situation might be to use the observed number of freckles on the second doll as the best estimate of the likely number of freckles on the outermost doll. Such an estimator will necessarily be biased, however, because the second doll is smaller than the outermost doll and necessarily possesses a smaller number of freckles. In other words, employing n1 as an estimate of n0 necessarily underestimates the true number of freckles on the outermost doll. This is where the bootstrapped estimator, n2, reveals its true value. Because the third doll is smaller than the second doll by an amount similar to that by which the second doll is smaller than the outermost doll, the ratio of the number of freckles on the two inner dolls, n1: n2, should be a close approximation of the ratio of the number of freckles on the second doll to number on the outer doll, n0: n1. This in a nutshell is the principle underlying the bootstrap procedure.

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