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Marginal curvatures and their usefulness in the analysis of nonlinear regression models. (English) Zbl 0623.62062
Interval estimation for individual parameters in nonlinear regression is examined in relation to curvature of the solution locus. Criteria are developed to define severe curvature and distinguish cases in which linear methods may be used from those in which parameter transformations are necessary. A formula for improving the linear approximation in cases of relatively mild curvature is proposed.
For example, suppose that interest is focused on one specific parameter, \(\theta_ j\), from among \(\theta =\{\theta_ 1,\theta_ 2,...\), \(\theta_ p\}\) in a nonlinear regression model. A crude approach to setting 95 % confidence limits to \(\theta_ j\) might proceed as follows: First calculate the standard error of the estimate [SE(\({\hat \theta}_ j)]\) as the square root of its asymptotic variance. Then call \({\hat \theta}_ j\pm 1.96 SE({\hat \theta}_ j)\) the confidence limits. In this article, higher-order correction terms \(\Gamma\) and \(\beta\) are proposed that enable us to develop a power series expansion for the confidence limits, as \[ {\hat \theta}_ j-1.96 SE({\hat \theta}_ j)\{1+(1.96)\Gamma {\hat \sigma}+(1.96)^ 2\beta {\hat \sigma}^ 2+...\} \] \[ and\quad {\hat \theta}_ j+1.96 SE({\hat \theta}_ j)\{1-(1.96)\Gamma {\hat \sigma}+(1.96)^ 2\beta {\hat \sigma}^ 2+...\}, \] respectively. Here \({\hat \sigma}^ 2\) denotes some estimate of \(\sigma^ 2\), the variance of a single observation.

62J02 General nonlinear regression
62F10 Point estimation
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