# Delta Method

The delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.

## Method

While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence of random variables $X_n$ satisfying

[$]{\sqrt{n}[X_n-\theta]\,\xrightarrow{D}\,\mathcal{N}(0,\sigma^2)},[$]

where $\theta$ and $\sigma^2$ are finite valued constants and $\xrightarrow{D}$ denotes convergence in distribution, then

[$] {\sqrt{n}[g(X_n)-g(\theta)]\,\xrightarrow{D}\,\mathcal{N}(0,\sigma^2[g'(\theta)]^2)} [$]

for any function $g$ satisfying the property that $g'(\theta)$ exists and is non-zero valued.

The method extends to the multivariate case. By definition, a consistent estimator $B$ converges in probability to its true value $\beta$, and often a central limit theorem can be applied to obtain asymptotic normality:

[$]\sqrt{n} (B-\beta )\,\xrightarrow{D}\,\mathcal{N}(0, \Sigma ),[$]

where n is the number of observations and $\Sigma$ is a covariance matrix. The multivariate delta method yields the following asymptotic property of a function $h$ of the estimator $B$ under the assumption that the gradient $\nabla h$ is non-zero:

[$]\sqrt{n}(h(B)-h(\beta))\,\xrightarrow{D}\,\mathcal{N}(0, \nabla h(\beta)^T \cdot \Sigma \cdot \nabla h(\beta)).[$]

## References

• Wikipedia contributors. "Delta method". Wikipedia. Wikipedia. Retrieved 30 May 2019.