Delta Method

The Big Idea: Linearizing the Non-Linear

The Delta Method is a powerful analytical tool in statistics that allows us 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, $Var(\hat \theta)$.

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 or very difficult due to the non-linear nature of $g$.

The Delta Method’s solution is simple: It replaces the complicated non-linear function with a simple linear one.

How it Works: The First-Order Taylor Approximation

The method’s foundation is the first-order Taylor Series approximation from calculus. This technique allows us to approximate a complex function $g(\hat \theta)$ near a known point $\theta$ with a tangent line:

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

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 $\theta$.
• $(\hat \theta − \theta)$ is the error, or the difference between our estimate and the true value.

Since the right side is a linear expression, 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, $a$).
• $\hat \theta−\theta$ 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.

• Slope effect: If the absolute value of the slope is small $(∣g′(\theta)∣<1)$, the transformation compresses the estimate’s variability, resulting in a smaller approximate variance. If the slope is large $(∣g′(\theta)∣>1)$, the variability is stretched, resulting in a larger approximate variance.

Asymptotic Theory and the Role of $\sqrt{n}$

The Delta Method is most powerful when viewed through the lens of asymptotic theory (the behavior of statistics as the sample size $n$ approaches infinity).

The Problem of Collapsing Variance

By the Law of Large Numbers, the difference between the estimate and the true value, $(\hat \theta − \theta)$, must converge to 0 as $n \rightarrow \infty$. This means the variance also collapses to 0. An estimate that always converges to a single point is useless for statistical inference (like creating confidence intervals).

To prevent this “collapse” and derive a meaningful, non-degenerate distribution, statisticians use a normalizing factor, which is where $\sqrt{n}$​ comes in.

The scaling factor $\sqrt{n}$ counteracts the $\frac{1}{n}$​ decrease in variance that is typical for sample means (whose variance is $\frac{\sigma^2}{n}$​) and other common estimators. By multiplying by $\sqrt{n}$​, we “magnify” the difference just enough to keep the variance from disappearing, allowing the distribution to converge to a non-degenerate, finite distribution (the Normal distribution). The constant $\sqrt{n}$ gets squared so it cancels out with the denominator $n$ of the estimate.

Application of $\sqrt{n}$ in the Delta Method

The method is typically applied to estimators that are root-n consistent, meaning their normalized error converges to a normal distribution, usually based on the Central Limit Theorem (CLT):

\[\sqrt{n} (\hat \theta_n - \theta) \xrightarrow {d} \mathcal {N} (0, Var)\]

where $Var$ is the asymptotic variance of $\hat \theta_n$

The Delta Method theorem formally proves that if the original estimator $\hat \theta_n$​ has this property, the transformed estimator $g(\hat \theta_n$) will also be asymptotically normal:

\[n​(g(\hat \theta_n​)−g(\theta)) \xrightarrow {d} \mathcal {N} (0,[g′(\theta)]^2⋅Var​)\]

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 $\theta$. In practice, 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.