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 x_{1}, . . ., x_{n} be data observed as a realization of the random variables X_{1}, . . ., X_{n} ~ 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 ...