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The signed root deviance profile and confidence intervals in maximum likelihood analysis. (English) Zbl 0882.62026

Summary: We investigate a natural extension of the profile \(t\) plot of D. M. Bates and D. G. Watts [Nonlinear regression analysis and its applications. (1988; Zbl 0728.62062)] to a general parametric function \(g(\theta)\) of the parameters \(\theta\) in a general maximum likelihood analysis. Although the basic purpose of the extension, called the signed root deviance profile (SRDP), is to construct likelihood ratio (LR) confidence intervals for \(g(\theta)\), it has various other applications that significantly extend its usefulness. The tangent to the plot of the SRDP at the maximum likelihood estimate \(\widehat g\) of \(g(\theta)\) gives the linear approximation (LA) interval based on the observed information matrix. The plot may be used as a diagnostic tool to compare LA and LR intervals and to suggest transformations of \(g(\theta)\) whose LA intervals when inverted are close to the LR intervals for \(g(\theta)\). The standard way to construct any profile is through repeated optimizations, but problems associated with nonlinear constraints can make this difficult. An alternative method based on integration is presented that avoids these problems. An example and a simulation study are given to illustrate the proposed methods.

MSC:

62F25 Parametric tolerance and confidence regions

Citations:

Zbl 0728.62062
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