In Auburngrads/teachingApps: Apps for Teaching Statistics, R Programming, and Shiny App Development

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) A parameter for which we know the variance - $\theta$

2) The variance of the paramter - $Var[\theta]$

2) A function of the parameter - $g(\theta)$

3) The partial derivatives - $\frac{\partial g(\theta)}{\partial \theta_{i}}, \;\; i=1,...m$

The General Delta Method Equation

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

Auburngrads/teachingApps documentation built on June 17, 2020, 4:57 a.m.