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### Understanding the Delta Method

- Suppose we know the variance of some estimator  $\mathbf{\hat{\theta}}=\hat{\theta}_{1},...,\hat{\theta}_{r}$

- But, we want to find the variance of some other estimator which is a function of $\mathbf{\theta}$, say $g(\mathbf{\hat{\theta}})$

- Example functions 

    + $g(\mathbf{\hat{\theta}})=\log[\theta]$

    + $g(\mathbf{\hat{\theta}})=\theta^2$

- The Delta Method can help us estimate $\widehat{Var}[g(\mathbf{\hat{\theta}})]$ from $\widehat{Var}[\mathbf{\hat{\theta}}]$ if we can find

$$\frac{dg(\theta)}{d\theta}$$

### The Delta Method Requires Us To Find/Compute 4 Things

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1) <red>__A parameter for which we know the variance - $\theta$__</red>

2) <purple>__The variance of the paramter - $Var[\theta]$__</purple>

2) <blue>__A function of the parameter - $g(\theta)$__</blue>

3) <green>__The partial derivatives - $\frac{\partial g(\theta)}{\partial \theta_{i}}, \;\; i=1,...m$__</green>


## The General Delta Method Equation

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<small>
$$ 
Var\left[g(\hat{\theta})\right]\approx \sum_{i=1}^{r} \left[\frac{\partial g(\mathbf{\theta})}{\partial \theta_{i}}\right]^{2} Var(\hat{\theta_{i}})+\sum_{i=1}^r \mathop{\sum^{r}_{j=1}}_{i\ne j}\left[\frac{\partial g(\mathbf{\theta})}{\partial \theta_{i}}\right]\left[\frac{\partial g(\mathbf{\theta})}{\partial \theta_{j}}\right] Cov(\hat{\theta}_{i}, \hat{\theta}_{j})
$$
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