Delta method

The Delta method is used to calculate the asymptotic variance of a random variable that is a function of another random variable. The derivative of the function and the mean and asymptotic variance of the second RV are used.

Let $Z_n$ be a sequence of random variables such that

$$\sqrt{n}(Z_n - \theta) \xrightarrow [n\to \infty ]{(d)} \mathcal{N}(0, \sigma^2)$$

where $\sigma^2$ is the asymptotic variance, and $\theta \in \mathbb{R}$. This means that $Z_n$ is asymptotically normal.

Given a function $g: \mathbb{R} \to \mathbb{R}$ that is continuously differentiable at $\theta$,

• $g(Z_n) \xrightarrow [n\to \infty ]{(\textbf{P})} g(\theta)$
• $(g(Z_n))_{n\geq 1}$ is also asymptotically normal with asymptotic variance $g'(\theta)^2\sigma^2$
• In other words,

$$\sqrt{n}(g(Z_n) - g(\theta)) \xrightarrow [n\to \infty ]{(d)} \mathcal{N}(0, g'(\theta)^2\sigma^2)$$