---
# System prepended metadata

title: CLT and Delta method

---

# CLT and Delta method
## CLT
The formal asymptotic equation,

$$
\sqrt{n} (\hat{\beta} - \beta) \sim_d N(0, V(\hat{\beta}))
$$

Informal asymptotic equation,

$$
(\hat{\beta} - \beta) \sim_d N(0, \frac{1}{n}V(\hat{\beta}))
$$

However, we mostly report $\hat{\beta}$ and estimated variance as

$$
\hat{\sigma^2} = \frac{1}{n}\widehat{V(\hat{\beta})}.
$$


## Delta method
The formal delta method,

$$
\sqrt{n} (h(\hat{\beta}) - h(\beta)) \sim_d N \left(0, h'(\beta)^T V(\hat{\beta}) h'(\beta) \right)
$$

Informal delta method,

$$
 (h(\hat{\beta}) - h(\beta)) \sim_d N \left (0, \frac{1}{n} h'(\beta)^T V(\hat{\beta}) h'(\beta) \right)
$$

The estimated variance for $h(\hat{\beta})$ should be

$$
 \frac{1}{n} h'(\hat{\beta})^T \widehat{V(\hat{\beta})} h'(\hat{\beta}) = h'(\hat{\beta})^T \hat{\sigma^2} h'(\hat{\beta})
$$

##  Why informal

Formally,

$$
(\hat{\beta} - \beta) \sim_d 0
$$ 

is a constant without variance.