Delta Method

Table of contents

1. Univariate Delta Method

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Univariate Delta Method

Let $X_n$ be a RV that asymptotically follows a normal distribution (see CLT).

\[X_n \leadsto N(\mu, \sigma^2)\]

If $g$ is a differentiable function and $g’(\mu) \neq 0$, then the distribution of $g(X_n)$ converges to a normal distribution:

$$ g(X_n) \leadsto N\left(g(\mu),\, (g'(\mu))^2\, \frac{\sigma^2}{n}\right) $$

Taylor Expansion

The first order Taylor expansion approximation of $g(X_n)$ around $\mu$ is:

\[g(X_n) \approx g(\mu) + g'(\mu) (X_n - \mu)\]

Since $g’(\mu) \neq 0$, rearranging the terms:

\[X_n - \mu \approx \frac{g(X_n) - g(\mu)}{g'(\mu)}\]

Multiplying both sides by $\frac{\sqrt{n}}{\sigma}$:

\[\frac{\sqrt{n} (X_n - \mu)}{\sigma} \approx \frac{\sqrt{n}(g(X_n) - g(\mu))}{g'(\mu) \sigma} \leadsto N(0, 1)\]

The standardized form makes it easy to see that

\[g(X_n) \leadsto N\left(g(\mu),\, (g'(\mu))^2\, \frac{\sigma^2}{n}\right)\]