The expected number of local maxima of a random field and the volume of tubes. (English) Zbl 0801.62087

The problems of statistical testing and confidence intervals for a nonlinearity in a linear model lead to that of evaluation of the volume of curvilinear tubes in the multidimensional unit sphere. The paper generalizes, for the multidimensional parameter model, the approach of I. Johnstone and D. Siegmund [Ann. Stat. 17, No. 1, 184-194 (1989; Zbl 0678.62066)], based on a probabilistic argument relating the tube’s volume estimator to that for the expectation of the number of local maxima of a random field. The result agrees, improves, or can be worse than the Hotelling-Naiman’s, depending on the dimension and tube’s radius.


62M40 Random fields; image analysis
62J02 General nonlinear regression


Zbl 0678.62066
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