Confidence regions in nonlinear regression and geometry

The goal of this thesis is to find confidence regions in nonlinear regession. The problem can be reduced to the approximation of a certain tail probability. The idea of the approximation is based on computing the expected numbers of upcrosings and local maxima of a random field.;When the regression model involves only one or two nonlinear parameters, we have both theoretical and numerical methods to derive conservative confidence regions. And the empirical evidence suggests that in practice the regions are essentially exact. Many interesting and useful models, such as semilinear regression, two-phase regression, and linear functional models, can be fit into this framework.;Commonly used confidence regions for parameter subsets in nonlinear regression are based upon an approximate F method that derives from the assumption that an obvious analogue of the usual F statistic of linear regression analysis has approximately an F distribution. Examples from literature are used to illustrate the new procedure and the approximate F method.;Finally, there is a need for more work when the regression model has more than two nonlinear parameters.