A confidence set for that x-coordinate where a quadratic regression model has a given gradient

摘要

An exact confidence set for that x-coordinate where a quadratic regression model has a given gradient is derived. The limits of the confidence set are given by mathematical formulae. They are implemented in Fortran programs that can be downloaded from the web. The confidence set need not be an interval. Its increase and its changing shape for increasing confidence level is extensively described and visualized in a figure that relates to data from nitrogen-rate trials in Germany. The wheat yields in this example are modeled as quadratic functions of the nitrogen input in order to determine a confidence set for the economically optimum nitrogen fertilization. The disadvantage that the confidence set does not distinguish between concave and convex parabolae, between profit maxima and minima, is also discussed.