Delta Method

The Big Idea: Linearizing the Non-Linear

The Delta Method is a powerful analytical tool in statistics used to find the approximate variance (or standard error) of a complicated, non-linear function of an estimator.

Imagine you have a reliable statistical estimate, $\hat \theta$, of a true population parameter, $\theta$. You also know the variance of this estimate, $Var(\hat \theta)$.

The Problem: Now, suppose your real interest lies in a transformed version of $\hat \theta$, represented by a non-linear function $g(\hat \theta)$ (e.g., $g(\hat \theta)=\log(\hat \theta)$ or $g(\hat \theta)=\theta^2$). Calculating the exact variance, $Var(g(\hat \theta))$, is often analytically impossible because non-linear functions warp distributions.

The Solution: The Delta Method’s solution is simple: It replaces the complicated non-linear function with a simple linear one by applying first-order Taylor Approximation.

Step 1: The Intuitive Approach (Taylor Approximation)

Recall that a Taylor Series approximates a complex function near a known point with a tangent line. In our case, we approximate our transformed estimator $g(\hat \theta)$ near the true value $\theta$:

\[g(\hat \theta) \approx g(\theta) + g'(\theta) (\hat \theta−\theta)\]

In this approximation:

• $g(\theta)$ is the value of the function at the true (but unknown) parameter $\theta$.
• $g’(\theta)$ is the slope (the first derivative) of the function $g$ evaluated at the true value.
• $(\hat \theta − \theta)$ is the error of our estimate.

If we subtract $g(\theta)$ from both sides, we get an equation that links the error of the output directly to the error of the input:

\[g(\hat \theta) - g(\theta) \approx g'(\theta) \cdot (\hat \theta−\theta)\]

This equation reveals the core mechanism: the error in our new metric is simply the original error multiplied by the slope $g’(\theta)$.

Since variance is just the average squared error, we can easily apply the standard rules for the variance of a linear transformation: $Var(aX+b)=a^{2}Var(X)$.

• $g(\theta)$ is a constant, so its variance is 0.
• $g’(\theta)$ is a fixed slope (a constant multiplier).
• $\hat \theta−\theta$ has the same variance as $\hat \theta$ (since subtracting a constant doesn’t change variance).

Applying the variance rules to the approximation yields the Delta Method formula for the approximate variance:

\[Var(g(\hat \theta)) \approx [g'(\theta)]^{2} \cdot Var(\hat \theta)\]

The approximate variance of the transformed statistic is the original variance, scaled by the square of the function’s derivative (slope) evaluated at the true value.

• If the absolute value of the slope is small $(∣g′(\theta)∣<1)$, the transformation compresses the variability.
• If the slope is large $(∣g′(\theta)∣>1)$, the transformation stretches the variability.

Step 2: The Formal Approach (Asymptotic Theory)

The derivation above is intuitive, but to use this for Hypothesis Tests or Confidence Intervals, we need to prove that the resulting distribution is actually Normal. This requires Asymptotic Theory (what happens as sample size $n$ approaches infinity).

The Problem of Collapsing Variance

By the Law of Large Numbers, our estimator $\hat \theta_n$​ gets closer to $\theta$ as $n$ increases. Consequently, the error $(\hat \theta_{n}−\theta)$ shrinks to zero. We can see this clearly with the sample mean $\bar X$. Its variance is:

\[\text{Var}(\bar X) = \frac{\sigma^2}{n}\]

As $n \rightarrow \infty$, the distribution collapses into a single point. A single point has no “shape,” so we cannot calculate probabilities or confidence intervals.

The Fix: Magnifying the Error

To study the shape of the distribution before it vanishes, we put it under a “microscope.” We scale the error up by $\sqrt{n}$​.

• The variance shrinks by $\frac{1}{n}$​.
• Multiplying the error by $\sqrt{n}$​ scales the variance by $(\sqrt{n})^2 = n$.
• This perfectly cancels the shrinkage, stabilizing the distribution so it converges to a non-degenerate Normal distribution.

The Formal Statement

We assume our original estimator satisfies the Central Limit Theorem. This theorem describes the distribution of the magnified error:

\[\sqrt{n}(\hat \theta_{n}−\theta) \xrightarrow {d} \mathcal {N} (0, \sigma^{2})\]

The Delta Method Theorem simply applies our linear approximation from Step 1 to this magnified error:

\[\sqrt{n}​(g(\hat \theta_n​)−g(\theta)) \xrightarrow {d} \mathcal {N} (0,[g'(\theta)]^2 \cdot \sigma^{2}​)\]

This confirms that the transformed error also follows a Normal distribution, with the variance scaled by the slope squared.

In practice we don’t know the true value of $\theta$ but we do have a consistent estimator $\hat \theta$, and as the sample grows it coverges to $\theta$. Therefore, we can plug $\hat \theta$ in place of $\theta$ when we calculate the derivative.

Why is It Useful

Once we have established the asymptotic normality and found the approximate variance of the transformed statistic, we can move from theoretical approximation to practical inference:

• Calculate the Standard Error (SE): The SE is simply the square root of the final approximate variance.
• Construct Confidence Intervals: Using the property of asymptotic normality, we can build large-sample confidence intervals for the transformed parameter $g(θ)$.
• Perform Hypothesis Tests: The approximate normal distribution allows us to calculate z-scores and p-values for transformed quantities.