Chapter 9
Bootstrapping
9.1 The Bootstrap Principle
Let θ = T(F) be an interesting feature of a distribution function, F, expressed as a functional of F. For example, T(F) = ∫ z dF(z) is the mean of the distribution. Let x1, . . ., xn be data observed as a realization of the random variables X1, . . ., Xn ~ i . i . d . F. In this chapter, we use X ~ F to denote that X is distributed with density function f having corresponding cumulative distribution function F. Let X={X_1,. . .,X_n} denote the entire dataset.
If 
is the empirical distribution function of the observed data, then an estimate of θ is 
. For example, when θ is a univariate population mean, the estimator is the sample mean, 
.
Statistical inference questions are usually posed in terms of 
or some 
, a statistical function of the data and their unknown distribution function F. For example, a general test statistic might be 
, where ...
