Many phenomena in the life sciences can be analyzed by using a fixed design regression model with a regression function m that exhibits a crossing-point in the following sense: the regression function runs below or above its mean level, respectively, according as the input variable lies to the left or to the right of that crossing-point, or vice versa. We propose a non-parametric estimator and show weak and strong consistency as long as the crossing-point is unique. It is defined as maximizing point arg max of a certain marked empirical process. For testing the hypothesis H ₀ that the regression function m actually is constant (no crossing-point), a decision rule is designed for the specific alternative H ₁ that m possesses a crossing-point. The pertaining test-statistic is the ratio max/argmax of the maximum value and the maximizing point of the marked empirical process. Under the hypothesis the ratio converges in distribution to the corresponding ratio of a reflected Brownian bridge, for which we derive the distribution function. The test is consistent on the whole alternative and superior to the corresponding Kolmogorov-Smirnov test, which is based only on the maximal value max. Some practical examples of possible applications are given where a certain study about dental phobia is discussed in more detail.