**APPENDIX 1**

**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

(A.1)

where

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

(A.2)

where *σ ^{2}* is the variance of X. The delta ...