Table 1 Simulation power in nine sample functions.

y = 0 + N(0,1)

0.050

0.058

0.053

0.055

0.068

0.059

0.053

0.046

y = x + N(0,1)

0.839

0.958

0.951

0.950

0.940

0.946

0.544

0.593

y = \(0.5({{\rm{x}}+1)}^{2}\) + N(0,1)

0.861

0.961

0.949

0.946

0.935

0.946

0.580

0.608

y = sin(\(\pi \) x) + N(0,1)

0.957

0.937

0.912

0.904

0.963

0.962

0.742

0.805

y = sin(3 \(\pi \) x) + N(0,1)

0.795

0.180

0.182

0.190

0.201

0.174

0.694

0.423

y = cos(\(\pi \) x) + N(0,1)

0.947

0.066

0.079

0.073

0.690

0.653

0.726

0.649

y = cos(2 \(\pi \) x) + N(0,1)

0.888

0.060

0.065

0.066

0.151

0.109

0.720

0.570

y = cos(3 \(\pi \) x) + N(0,1)

0.707

0.064

0.072

0.070

0.109

0.093

0.688

0.394