Resampling Methods: Bootstrap and Jackknife

Why Resampling Methods Matter

Traditional statistical inference relies on analytical formulas for standard errors and confidence intervals, which often assume normality or require large samples. Resampling methods bypass these assumptions by using the data itself to estimate the sampling distribution of a statistic. This makes them especially valuable in actuarial work, where data is often skewed, heavy-tailed, or too small for asymptotic results to be reliable.

The two main resampling approaches are the bootstrap and the jackknife. The bootstrap is more flexible and powerful, while the jackknife is computationally simpler and provides quick bias and variance estimates. Both methods are tested on Exam MAS-I.

The Bootstrap Algorithm

The nonparametric bootstrap works as follows. From an original sample of n observations x_1, x_2, ..., x_n, draw B bootstrap samples, each of size n, by sampling with replacement from the original data. Some observations will appear more than once in a bootstrap sample, and others will not appear at all. For each bootstrap sample b = 1, 2, ..., B, compute the statistic of interest theta_hat_b. The collection of B bootstrap statistics approximates the sampling distribution of the statistic.

For example, suppose you have 8 claim amounts: 120, 350, 200, 180, 500, 150, 280, 900. The sample mean is 335. One bootstrap sample might be {200, 900, 350, 200, 120, 280, 350, 500}, giving a bootstrap mean of 362.5. Another might be {150, 180, 120, 350, 280, 200, 150, 120}, giving a bootstrap mean of 193.75. After generating B = 1000 or more bootstrap means, the standard deviation of these bootstrap means estimates the standard error of the sample mean.

Bootstrap Confidence Intervals

The percentile method constructs a confidence interval by taking the alpha/2 and 1 - alpha/2 quantiles of the B bootstrap statistics. For a 95% confidence interval, use the 2.5th and 97.5th percentiles. If the 1000 bootstrap means range from 180 to 520 with the 2.5th percentile at 215 and the 97.5th percentile at 475, the 95% percentile bootstrap confidence interval is (215, 475).

The basic bootstrap interval uses a different construction: (2*theta_hat - q_{1-alpha/2}, 2*theta_hat - q_{alpha/2}), where q denotes bootstrap quantiles and theta_hat is the original sample estimate. This interval reflects the bootstrap distribution around the original estimate rather than using the percentiles directly. The bias-corrected and accelerated (BCa) method adjusts for both bias and skewness in the bootstrap distribution, producing more accurate coverage probabilities, especially for small samples or skewed statistics.

Parametric Bootstrap

The parametric bootstrap generates samples from a fitted parametric model rather than resampling the data. First, fit a parametric distribution (such as a lognormal) to the original data, obtaining parameter estimates. Then generate B samples of size n from the fitted distribution, compute the statistic for each, and use these to estimate standard errors and confidence intervals.

The parametric bootstrap is more efficient than the nonparametric bootstrap when the parametric model is correct, because it uses the model structure to generate more representative samples. However, if the assumed model is wrong, the parametric bootstrap can give misleading results. In actuarial applications, the parametric bootstrap is commonly used for constructing confidence intervals around fitted loss distribution parameters.

The Jackknife Method

The jackknife systematically removes one observation at a time from a sample of n observations. For each i = 1, 2, ..., n, compute the leave-one-out estimate theta_hat_{(i)} using the remaining n - 1 observations. The jackknife estimate of bias is bias_jack = (n - 1) * (theta_bar_{(.)} - theta_hat), where theta_bar_{(.)} = (1/n) * sum(theta_hat_{(i)}) is the average of the leave-one-out estimates and theta_hat is the estimate from the full sample.

The jackknife variance estimate is Var_jack = ((n - 1) / n) * sum((theta_hat_{(i)} - theta_bar_{(.)})^2). This formula includes the factor (n - 1)/n rather than the more typical 1/(n - 1) because the leave-one-out estimates are highly correlated with each other, and the inflation factor compensates for this correlation.

Jackknife Worked Example

Suppose 5 loss observations are 100, 200, 300, 400, 1000, and you want to estimate the standard error of the sample mean (which is 400). The leave-one-out means are: theta_hat_{(1)} = (200+300+400+1000)/4 = 475, theta_hat_{(2)} = 500/4 + 1000/4 = 450, theta_hat_{(3)} = (100+200+400+1000)/4 = 425, theta_hat_{(4)} = (100+200+300+1000)/4 = 400, theta_hat_{(5)} = (100+200+300+400)/4 = 250. The average is theta_bar_{(.)} = (475+450+425+400+250)/5 = 400.

The jackknife variance is (4/5) * [(475-400)^2 + (450-400)^2 + (425-400)^2 + (400-400)^2 + (250-400)^2] = 0.8 * [5625 + 2500 + 625 + 0 + 22500] = 0.8 * 31250 = 25000. The jackknife standard error is sqrt(25000) = 158.1. This captures the large variability caused by the extreme observation of 1000.

Comparing Bootstrap and Jackknife

The bootstrap is more flexible and generally more accurate than the jackknife. It works well for virtually any statistic, including non-smooth statistics like the median and quantiles. The jackknife, by contrast, performs poorly for non-smooth statistics because removing a single observation can cause discrete jumps in the estimate. For the median of 5 observations, each leave-one-out estimate is one of only a few possible values, making the jackknife variance estimate unreliable.

The jackknife has the advantage of being deterministic (no random seed) and computationally simpler, requiring only n resamples versus B (typically 1000 or more) for the bootstrap. It also provides a straightforward bias estimate. In practice, the bootstrap has largely replaced the jackknife for confidence interval construction, but the jackknife remains useful for quick bias and influence diagnostics.

Common Pitfalls

A common mistake with the bootstrap is using too few bootstrap samples. While B = 100 may suffice for estimating standard errors, constructing reliable confidence intervals typically requires B = 1000 or more. Another error is applying the nonparametric bootstrap to dependent data (such as time series) without modification; block bootstrap or other specialized methods are needed for dependent data.

For the jackknife, students often forget the (n-1)/n factor in the variance formula or confuse it with the standard 1/(n-1) factor from sample variance. The jackknife variance formula is specific to the leave-one-out resampling scheme and differs from ordinary variance estimation.

Study Tips

Exam MAS-I tests your ability to implement both methods on small data sets and to understand their properties. Practice computing jackknife estimates by hand for samples of size 4 or 5. Know the bootstrap confidence interval constructions (percentile and basic methods) and understand when the parametric bootstrap is preferred over the nonparametric version. Be prepared to compare the strengths and weaknesses of bootstrap versus jackknife, and know that the jackknife fails for non-smooth statistics like the median.