The expected number of local maxima of a random field and the volume of tubes (Q1317258)

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 \textit{I. Johnstone} and \textit{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.