Simultaneous confidence bands for contrasts between several nonlinear regression curves

We propose simultaneous confidence bands of hyperbolic-type for the contrasts between several nonlinear (curvilinear) regression curves. Critical value of the confidence band is determined from the distribution of the maximum of a chi-square random process defined on the domain of explanatory variables. By means of the volume-of-tube method, we derive an upper probability formula of the maximum of a chi-square random process, which is sufficiently accurate in moderate tail regions. Moreover, we prove that the formula obtained is equivalent to the expectation of the Euler-Poincar\'e characteristic of the excursion set of the chi-square random process, and hence conservative. This result is regarded as a generalization of Naiman's inequality for Gaussian random processes. As an illustrative example, growth curves of consomic mice are analyzed.