The bootstrap

Idea: An error bar with no formula

The bootstrap treats the sample as the population and draws from it.

• Works for medians, Sharpe ratios, quantiles
• $30$ skewed fills have no usable textbook formula

Idea: The recipe

Draw $n$ points with replacement, compute $T$, repeat $B$ times.

• Some points repeat, some drop out
• The spread of $T_1$, …, $T_B$ estimates the standard error
• Take $B$ in the thousands

Example: Every resample by hand

Data $2$, $4$, $9$. What is the bootstrap variance of the sample mean?

1. Mean $5$, squared deviations $9+1+16=26$
2. Variance of the data: $\frac{26}{3}$
3. Mean of $3$ draws: $\frac{\frac{26}{3}}{3}=\frac{26}{9}$

Formula: What a resample misses

$P(\text{point}\text{left}\text{out})={\left(1-\frac{1}{n}\right)}^n$

Tends to 1/e ≈ 0.37, so a resample holds about 63% of the distinct points.

Quick check: A missing point

With $n=5$, what is the chance that a given point is missing from a resample?

Common mistake: Bootstrapping the maximum

The resample max equals the sample max about $63\%$ of the time. No value outside the sample can appear.

Instead Use it for means, medians and other statistics that move smoothly with the data.

Idea: When order matters

Plain resampling shuffles away runs of good and bad days.

• Returns $3$, $5$, $-4$, $-2$, $1$, $3$: plain gives $6\times \frac{58}{6}=58$
• Blocks of $2$ keep neighbours together
• Block length: hold the dependence, leave many blocks

Example: A block bootstrap

Returns in order: $3$, $5$, $-4$, $-2$, $1$, $3$. Resample $3$ blocks of $2$. What is the variance of the total?

1. Block totals $8$, $-6$, $4$, mean $2$
2. Squared deviations: $36+64+4=104$
3. Total of $3$ blocks: $3\times \frac{104}{3}=104$

Common mistake: Trusting a calm sample

If all $30$ fills came from a calm week, every resample is calm and the interval looks tight.

Instead The bootstrap measures the spread of an estimate. It does not repair a bad sample.