On the choice of difference quotients for evaluating prediction intervals

Abstract

The use of the Gauss approximation formula for prediction intervals of a function of several variables includes the determination of the partial derivatives. In many cases there may be difficulties to find these derivatives and they need to be replaced by difference quotients. The natural choice of step size in such quotients is to make it as small as possible that, however, may lead to numerical problems. Here, it is demonstrated that there is usually no reason for choosing small step sizes. Instead the best choice with respect to linearisation errors is proportional to the standard deviations of the variables involved. It is recommended that the difference quotient should be used with the step size equal to two standard deviations for an approximate 95% prediction interval.