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Estimation and testing of crossing‐points in fixed design regression
Author(s) -
Ferger Dietmar,
Klotsche Jens,
Lüken Ulrike
Publication year - 2012
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.2012.00521.x
Subject(s) - mathematics , estimator , empirical distribution function , statistics , brownian bridge , consistency (knowledge bases) , test statistic , regression analysis , p value , statistical hypothesis testing , brownian motion , discrete mathematics
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.