The Mean Value Theorem and Taylor's Expansion in Statistics

Themean value theorem and Taylor's expansion are powerful tools in statistics that are used to derive estimators from nonlinear estimating equations and to study the asymptotic properties of the resulting estimators. However, the mean value theorem for a vector-valued differentiable function does not exist. Our survey shows that this nonexistent theorem has been used for a long time in statistical literature to derive the asymptotic properties of estimators and is still being used. We review several frequently cited papers and monographs that have misused this theorem and discuss the flaws in these applications. We also offer methods to fix such errors.