Delta method

In statistics, 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.

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

where θ and σ² are finite valued constants and ${\displaystyle {\xrightarrow {D}}}$ denotes convergence in distribution, then

for any function g satisfying the property that g′(θ) exists and is non-zero valued.

Demonstration of this result is fairly straightforward under the assumption that g′(θ) is continuous. To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):

where ${\displaystyle {\tilde {\theta }}}$ lies between X_n and θ. Note that since ${\displaystyle X_{n}\,{\xrightarrow {P}}\,\theta }$ and ${\displaystyle X_{n}<{\tilde {\theta }}<\theta }$, it must be that ${\displaystyle {\tilde {\theta }}\,{\xrightarrow {P}}\,\theta }$ and since g′(θ) is continuous, applying the continuous mapping theorem yields

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