The Delta Method
Deriving an expression for an estimator of the variance of the estimator is one problem faced by statisticians when developing an estimator of a parameter. Both estimators are needed for confidence interval estimation and/or hypothesis testing.
Statisticians use a procedure commonly called the delta method to obtain an estimator of the variance when the estimator is not a simple sum of observations. The basic idea is to use a method from calculus called a Taylor series expansion to derive a linear function that approximates the more complicated function. We refer the reader to any introductory calculus text for a discussion of the Taylor series expansion.
To apply the delta method, the function must be one that can be approximated by a Taylor series and, in general, this means that it is a “smooth” function, with no “corners.” Consider such a function of a random variable X denoted as ƒ (X). To apply the delta method, we use the first two terms of a Taylor series expansion about the mean of the variable to approximate the value of the function as
is the derivative of the function with respect to X evaluated at the mean of X. It follows from (A.l) that the variance of the function is approximately
where σ2 is the variance of X. The delta ...