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