---
tags: metric, memo, public
---
# Delta Method
## Delta Method (first order)

Let $Y_n$ be a sequence of random variables. For a given function $g$ and a specific value of $\theta$, suppose that $g'(\theta)$ exists and is not $0$. If

$$
\sqrt{n} (Y_n - \theta) \to^d N(0, \sigma^2),
$$

then

$$
\sqrt{n} [g(Y_n) - g(\theta)] \to^d N(0, \sigma^2[g'(\theta)]^2).
$$

## Delta Method (second order)

Let $Y_n$ be a sequence of random variables. For a given function $g$ and a specific value of $\theta$, suppose that $g'(\theta)=0$, $g''(\theta)$ exists and is not $0$. If

$$
\sqrt{n} (Y_n - \theta) \to^d N(0, \sigma^2),
$$

then

$$
n [g(Y_n) - g(\theta)] \to^d \sigma^2 \frac{g''(\theta)}{2} \chi^2_1 .
$$

## Delta Method (Denis' version)

Suppose that $X_n \to^d X$ and $g(·)$ is differentiable at $a$, $b_n \to 0$. Then

$$
\frac{g(a + b_n X_n) - g(a)}{b_n} \to^d X g'(a).
$$

If $g′(a) = 0$ and $g′′(a)$ exists, then

$$
\frac{g(a + b_n X_n) - g(a)}{b_n^2} \to^d \frac{1}{2} X^2 g''(a).
$$