Comparison of feature significance quantile approximations

Curve estimates and surface estimates often contain features such as inclines, bumps or ridges which may signify an underlying structural mechanism. However, spurious features are also a common occurrence and it is important to identify those features that are statistically significant. A method has been developed recently for recognising feature significance based on the derivatives of the function estimate. It requires simultaneous confidence intervals and tests, which in turn require quantiles for the maximal deviation statistics. This paper reviews and compares various approximations to these quantiles. Applying upcrossing-probability theory to this problem yields better quantile approximations than the use of an independent blocks method.