Resampling Methods: Bootstrap and Jackknife
Apply bootstrap and jackknife resampling methods for statistical inference on Exam MAS-I.
Bootstrap Methods
The bootstrap estimates the sampling distribution of a statistic by resampling with replacement from the observed data. From the original sample of size n, draw B bootstrap samples (each of size n with replacement), compute the statistic for each, and use the B bootstrap statistics to estimate standard errors, confidence intervals, and bias. The percentile method uses the alpha/2 and 1-alpha/2 quantiles of bootstrap statistics as confidence limits. The bias-corrected and accelerated (BCa) method adjusts for bias and skewness. Parametric bootstrap generates samples from a fitted parametric model instead of resampling data.
Jackknife Method
The jackknife systematically omits one observation at a time. From n observations, compute n leave-one-out estimates theta-hat_{(i)}. The jackknife estimate of bias is (n-1)*(theta-bar_{(.)} minus theta-hat). The jackknife variance estimate is ((n-1)/n) * sum of (theta-hat_{(i)} minus theta-bar_{(.)})^2. The jackknife is computationally simpler than bootstrap (only n resamples) but less flexible. It works well for smooth statistics but poorly for non-smooth ones like the median. Exam MAS-I tests both methods and their applications to actuarial data.