Delta Method Transformations

Worked Example

Consider a Bernoulli(0.5) sample of size 10, so the sample mean K/n has expected value p=0.5 and variance p(1-p)/n = 0.25/10 = 0.025. Let the transformation be g(x) = x².

1. Evaluate the derivative at p: g'(x) = 2x, so g'(0.5) = 1.0.
2. Calculate the first-order delta variance approximation: [g'(p)]² · Var(K/n) = 1.0² · 0.025 = 0.025.
3. Calculate the exact transformed expectation:
   E[(K/n)²] = Var(K/n) + (E[K/n])² = 0.025 + 0.5² = 0.025 + 0.25 = 0.275.
4. The exact variance of (K/n)² for n=10 at p=0.5 can be calculated from the higher moments of the binomial distribution, yielding 0.026125.

Notice that E[g(mean)] = 0.275 is strictly greater than g(E[mean]) = 0.25, an example of Jensen's inequality for a convex function.

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.

Limitations

The first-order delta method provides a normal approximation that is asymptotically valid as n → ∞. Do not call an asymptotic approximation an equality. For small n (like in our finite domain up to 20), or when the derivative vanishes (as in the centered square at p=0.5), the approximation fails to capture the true behavior. The true exact distribution of a transformed binomial proportion is discrete and has bounded support, unlike the unbounded continuous normal approximation implied by the delta method variance.

References

The conditions and proof for the delta method can be found at StatLect: Delta Method. This resource explains the general conditions required for the Taylor series approximation to hold, and the worked examples and boundary checks in this module demonstrate these principles in a finite binomial setting.