Confidence intervals and the bootstrap

How precise is an estimate

A point estimate such as a mean return says nothing about its reliability. A confidence interval adds it: for a mean, $\bar x\pm t^{*}\,s/\sqrt{n}$, where $t^{*}$ is the critical value of the $t$ distribution with $n-1$ degrees of freedom (about 2.06 for 95% and $n=25$, tending to 1.96 for large $n$). For a proportion the interval is $\hat p\pm1.96\sqrt{\hat p(1-\hat p)/n}$. The width is set by the standard error and the confidence level, and it shrinks only like $1/\sqrt{n}$.

Many statistics have no simple standard-error formula: a median, a Sharpe ratio, a drawdown, a ratio of two means. The bootstrap gets around that. Resample the observed data with replacement to make a new sample of the same size, compute the statistic on it, repeat thousands of times, and read the interval off the percentiles of the resulting values. It treats the sample as a stand-in for the population. It needs care when observations are dependent (resample blocks of consecutive days, not single days) and it cannot rescue a tiny sample.

• Confidence interval for a mean: estimate $\pm$ critical value $\times$ standard error.
• Bootstrap: resample with replacement, recompute, repeat, take percentiles; no formula needed.
• Dependent data: resample blocks of consecutive days so that autocorrelation survives.

Lab · what's graded

• An interval for a mean15 pts
• An interval for a proportion15 pts
• The steps of a bootstrap15 pts
• What 95% means15 pts
• Facts about the bootstrap15 pts
• Reading the percentile10 pts
• Why the bootstrap needs no formula15 pts