The Delta Method

If has a limiting Normal distribution, then the delta method allows us to find the limiting distribution of where is any smooth function.

Theorem (The Delta Method)

Suppose that

and that is a differentiable function such that . Then,

In other words,

Example

Let be IID with finite mean and finite variance . By the central limit theorem, . Let . Thus, where . Since , the delta method implies that

There is also a multivariant version of the Delta Method.

Theorem (The Multivariate Delta Method)

Suppose that is a sequence of random vectors such that

Let and let

Let denote evaluated at and assume that the elements of are nonzero. Then,

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 5.5.