All simulators
Delta Method vs Simulation
Push a normal estimate through ln, √, 1/x or x² and compare the delta-method normal with the truth.
Day 12: The Delta Methodsimulated g(x̄)delta-method approximation
Delta-method SE
0.2000
|g′(μ)| × SE
Simulated SD
0.2108
g(μ)
1.6094
Mean of g(x̄)
1.5912
gap = second-order bias
The estimate x̄ ~ N(μ, SE²) is pushed through g. Bars: the true distribution of g(x̄). Orange: the delta-method normal N(g(μ), (g′(μ)·SE)²). They agree when SE is small relative to the curvature of g.