Bootstrap Confidence Intervals

Every interval built so far needed something clean to invert: an exact pivot with a tabulated distribution (Pivotal Quantities), or the MLE's asymptotic normality (Wald and Likelihood-Ratio Intervals), or a normal-theory two-sample distribution (Two-Sample Tests). A Sharpe ratio, a median, a skewness coefficient, the difference of two medians — these have no clean closed-form sampling distribution, and often no obvious pivot at all. The bootstrap sidesteps the problem entirely: rather than deriving the sampling distribution of an estimator analytically, simulate it, by resampling from the data itself as a stand-in for the unknown population. This note builds the bootstrap principle from the same law-of-large-numbers idea that already justified Monte Carlo simulation, then constructs three interval methods — normal, pivotal (basic), and percentile — states the more refined BCa correction, and closes with the paradigm case a desk actually faces: a Sharpe ratio's sampling distribution has no textbook closed form, and the bootstrap is the standard way to attach a confidence interval to it.

Learning objectives. By the end of this note you should be able to: