Topics in U-statistics and Risk Estimation

I In the following, we compared the proposed unbiased variance estimator, V̂u = Q(m)−Q(0), with nonparametric bootstrap and smoothed bootstrap variance estimators. Table 1: Risk Based on L2 Loss True Unbiased Est Nonpar. Smoothed (simulated) V̂u V̂ar (UL2) 0.000464 0.000467 0.000499 0.000493 SD{V̂ar (UL2)} 1.525e-4 1.417e-4 1.307e-4 Bias/SD 0.019 0.247 0.222 Compute Time 40.76 hr 2.88 hr 4.47 hr Table 2: Risk Based on KL Loss B = True Unbiased Est Nonpar. Smoothed 1,000 (simulated) V̂u V̂ar (UKL) 0.005114 0.004919 0.004306 0.004364 SD{V̂ar (UKL)} 0.002258 0.001105 0.001104 Bias/SD 0.086 0.731 0.680 Compute Time 0.018 hr 8.40 hr 8.58 hr