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**Approximate confidence limits for a parameter function in nonlinear regression.**
*(English)*
Zbl 0607.62078

A two-step procedure is proposed for calculating approximate confidence limits for some specified function of the parameters of a nonlinear regression model having normal independent errors with constant variance \(\sigma^ 2.\)

If \(\eta\) (\(\theta\),x) denotes the regression function at a given value of the independent variable x and \(\theta\) is of dimension p, the first step involves a search for p ”transform points”, \(u_ 1,u_ 2,...,u_ p\) such that \(\phi_ i=\eta (\theta,u_ i)\) for \(i=1\) to p, defines a transformation from \(\theta\) to \(\phi\) with the log-likelihood nearly quadratic in \(\phi\). A simple method is to choose \(u_ i\) so that the corresponding \({\hat \phi}_ i\) are unbiased. The formulas needed to achieve this and alternative procedures are given in Section 2.

Suppose that the parametric function of interest is g(\(\theta)\). The second step now involves a search for the minimum and maximum values of g(\(\theta)\) on the surface of the ellipsoid in \(\phi\) space, which defines the boundary of the log-likelihood confidence region. This is achieved by an iterative algorithm described in Section 3.

Examples of the application are given for setting confidence limits to the asymptote and predicted values from the Mitcherlitz equation and for the point of infection and absolute growth rates for the Richards function growth curve. Further analysis is given in Section 5, illustrating the advantages of the proposed method over a direct Newton- Raphson-based search without transformation of parameters. It is shown that for given computational effort, the direct search is less reliable. The proposed method may also be used for the rapid and stable computation of a general profile likelihood.

If \(\eta\) (\(\theta\),x) denotes the regression function at a given value of the independent variable x and \(\theta\) is of dimension p, the first step involves a search for p ”transform points”, \(u_ 1,u_ 2,...,u_ p\) such that \(\phi_ i=\eta (\theta,u_ i)\) for \(i=1\) to p, defines a transformation from \(\theta\) to \(\phi\) with the log-likelihood nearly quadratic in \(\phi\). A simple method is to choose \(u_ i\) so that the corresponding \({\hat \phi}_ i\) are unbiased. The formulas needed to achieve this and alternative procedures are given in Section 2.

Suppose that the parametric function of interest is g(\(\theta)\). The second step now involves a search for the minimum and maximum values of g(\(\theta)\) on the surface of the ellipsoid in \(\phi\) space, which defines the boundary of the log-likelihood confidence region. This is achieved by an iterative algorithm described in Section 3.

Examples of the application are given for setting confidence limits to the asymptote and predicted values from the Mitcherlitz equation and for the point of infection and absolute growth rates for the Richards function growth curve. Further analysis is given in Section 5, illustrating the advantages of the proposed method over a direct Newton- Raphson-based search without transformation of parameters. It is shown that for given computational effort, the direct search is less reliable. The proposed method may also be used for the rapid and stable computation of a general profile likelihood.

### MSC:

62J02 | General nonlinear regression |

62F25 | Parametric tolerance and confidence regions |

65C99 | Probabilistic methods, stochastic differential equations |