Confidence Regions in Non-Linear Estimation

Summary The statistical properties of the approximate confidence regions for nonlinear estimation based on the likelihood ratio criterion are considered, assuming normal variation of errors. It is suggested that these approximate confidence regions are preferable to any known exact confidence regions for this problem, provided that the model is only moderately non-linear. A numerical measure of the non-linearity of a model (as a function of the experimental design and the parameter values) is proposed. This can be used to indicate: (a)the accuracy of these approximate confidence regions, and a simple method of modifying the regions to avoid exaggerating the significance of the experimental data in doubtful cases; and(b)the effectiveness of a non-linear transformation of parameters as a device for making the model approximately linear. Methods of computing the non-linearity so defined, either empirically or theoretically, are indicated.