Optimistic and pessimistic approximations to variance of time delay estimators

A general theorem is presented that can be used to find an optimistic or, if preferred, pessimistic expression for the variance of a time delay estimator. The theorem concerns the relative locations of the maxima of two functions which are contaminated with identical additive noise. The utility of the method developed is demonstrated by applying the approach to the generalized cross correlator to obtain both an optimistic expression and a pessimistic expression for the variance of the time delay.<>