Localized Sup-Norm Risk Bounds for Other Estimators

Empirical evaluation of sup-norm risk bounds for NW Kernel, Local Polynomial, Wavelet, and Spline estimators across regularity regimes.

0.049
Best Ratio (NW Kernel)
0.384
Max Ratio (all)
4
Estimator Classes
6
Bandwidth Settings
3
Sample Sizes

Empirical-to-Theoretical Ratio by Estimator (n=200)

Empirical Risk vs Bandwidth (n=200)

Ratio Scaling with Sample Size (h=0.1)

Theoretical Bound vs Empirical Risk (n=200, h=0.1)

Estimator Comparison (n=200, selected bandwidths)

Estimatorh=0.02 Ratioh=0.05 Ratioh=0.1 Ratioh=0.2 Ratioh=0.3 Ratioh=0.5 Ratio
NW Kernel0.0490.0940.1180.1380.1540.189
Local Poly0.0490.0940.1170.1380.1530.189
Wavelet0.2750.3020.3030.2780.2570.204
Spline0.1500.1760.2000.2370.2740.384

Key Findings

FindingDetails
Best PerformanceNW Kernel and Local Poly achieve ratios below 0.05 at small bandwidths
Bounds ValidityAll empirical risks remain well below theoretical bounds (ratio < 0.384)
Regime BehaviorKernel methods excel at small h (r<h regime); splines degrade at large h
Wavelet BehaviorWavelets show relatively stable ratios across bandwidths (~0.2-0.3)