Bootstrap

Study guides: Exam MAS-I, Exam MAS-II

The bootstrap estimates the sampling variability of a statistic by resampling the observed data with replacement B times and looking at the spread of the statistic across the resamples. It needs no distributional assumption and no formula for the standard error.

\mathrm{SE}_B(\hat{\alpha}) = \sqrt{\frac{1}{B-1}\sum_{r=1}^{B}\left(\hat{\alpha}^{*r} - \bar{\hat{\alpha}}^{*}\right)^{2}}

• Each bootstrap sample has the same size n as the original, drawn with replacement — so some observations appear several times and others not at all
• The probability a given observation misses a particular resample is \left(1 - \tfrac1n\right)^n \to e^{-1} \approx 0.368, so each resample uses about 63.2% of the distinct observations. Those left out are the out-of-bag set
• It is the resampling engine underneath Bagging and Random Forest
• Distinct from Cross-Validation: the bootstrap measures how variable an estimate is, cross-validation measures how well a model predicts
• Not the reserving bootstrap of Exam 7: the ODP Bootstrap Model samples with replacement from the Pearson residuals of an over-dispersed Poisson model fitted to a loss triangle and builds pseudo triangles from them, rather than resampling the observations themselves

Bootstrapping a Tail Quantile An actuary needs a standard error for the 95th percentile of a 250-claim severity sample. No parametric fit is assumed. How is it obtained, and roughly how many claims does each resample actually use?

Answer Draw B = 1{,}000 samples of 250 claims with replacement, compute the 95th percentile of each, and take the standard deviation of those 1,000 values — that is \mathrm{SE}_B. Each resample contains about 0.632(250) \approx 158 distinct claims; the remaining ~92 are duplicates of ones already drawn.