A Delta Method for Implicitly Defined Random Variables

Abstract

If random variables in one set are defined as explicit functions of random variables in a second set, Taylor series expansion (the delta method) may be used to prove the asymptotic normality of the first set of variates, under appropriate conditions, and to develop needed covariance estimates. Similar results are obtained for a set of random variables that are defined implicitly as functions of a second set of variables. This approach is used to calculate the variance of the attributable risk from case-control data.

Key Words:

• Asymptotic methods
• Asymptotic normality
• Attributable risk
• Implicit function
• Taylor series variance calculations