Delta Method Transformations

The delta method approximates the sampling distribution of a transformed random variable. If a sequence of random variables (like a sample mean) converges in probability to a constant, and a function is differentiable at that constant, we can approximate the variance of the transformed variable using the first-order Taylor expansion (the tangent line).

Experiment

Follow these steps using the Delta Method Transformations module to explore the limits of the first-order approximation:

1. Set Transform to Square, Success probability to 0.5, and Sample count to 10. Observe the Delta variance matches our calculated 0.025.
2. Change Transform to Centered square (which is g(x) = (x-0.5)²).
3. Observe that the Transform derivative at p=0.5 is exactly 0, meaning the Delta variance approximation is 0.
4. However, the Exact transformed variance is strictly positive (0.001125). This demonstrates that when the first derivative vanishes, higher-order terms dominate the true variance.