What is jackknife estimation, and how does it reduce bias compared to standard plug-in estimators?

The jackknife is a resampling technique that systematically removes one observation at a time to detect and correct bias in statistical estimators. While bootstrapping uses random resampling, the jackknife is deterministic and produces exactly n leave-one-out estimates from a sample of size n.\n\nAlgorithm:\n\n1. Compute the full-sample statistic theta_hat from all n observations\n2. For each i = 1 to n, remove observation i and compute theta_hat_{-i}\n3. The jackknife estimate of bias is: bias_J = (n - 1) x (mean of theta_hat_{-i} - theta_hat)\n4. The bias-corrected estimate is: theta_J = theta_hat - bias_J\n\nJackknife Standard Error:\n\nSE_J = sqrt[((n-1)/n) x sum of (theta_hat_{-i} - mean of theta_hat_{-i})^2]\n\nWorked Example:\n\nAnalyst Marco at Clearfield Research estimates the Sharpe ratio for a portfolio using 8 quarterly returns: {4.2%, 1.8%, -2.1%, 3.6%, 5.0%, -0.7%, 2.9%, 3.3%}.\n\nFull-sample Sharpe (annualized): mean = 2.25%, std = 2.41%, Sharpe = (2.25% / 2.41%) x sqrt(4) = 1.868\n\nLeave-one-out Sharpe ratios:\n\n| Dropped Obs | Remaining Mean | Remaining Std | Sharpe_{-i} |\n|---|---|---|---|\n| Q1 (4.2%) | 2.26% | 2.53% | 1.786 |\n| Q2 (1.8%) | 2.31% | 2.56% | 1.806 |\n| Q3 (-2.1%) | 2.87% | 1.72% | 3.337 |\n| Q4 (3.6%) | 2.06% | 2.55% | 1.614 |\n| Q5 (5.0%) | 1.86% | 2.14% | 1.738 |\n| Q6 (-0.7%) | 2.67% | 1.90% | 2.811 |\n| Q7 (2.9%) | 2.16% | 2.59% | 1.668 |\n| Q8 (3.3%) | 1.96% | 2.56% | 1.531 |\n\nMean of leave-one-out Sharpes: 2.036\n\nBias_J = (8 - 1) x (2.036 - 1.868) = 7 x 0.168 = 1.176\n\nBias-corrected Sharpe: 1.868 - 1.176 = 0.692\n\nThis reveals substantial upward bias in the small-sample Sharpe ratio, a well-documented phenomenon.\n\nJackknife vs. Bootstrap:\n- Jackknife is deterministic (no random seed issues) and computationally lighter\n- Bootstrap is more flexible and handles non-smooth statistics (medians, quantiles) better\n- Jackknife fails for statistics that are not smooth functions of the data\n- Bootstrap can produce confidence intervals with better coverage properties\n\nFor most CFA exam purposes, know that the jackknife is best for bias detection and correction, while bootstrapping excels at standard error and confidence interval construction.\n\nDive deeper into resampling methods in our CFA Quantitative Methods course.