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Bootstrap confidence intervals for a class of parametric problems. (English) Zbl 0567.62025
The paper considers an application of the bootstrap method to obtain confidence intervals for $\theta =t(\eta)$ on the basis of data y coming from a multivariate normal with mean $\eta$ and variance-covariance matrix identity. The usual approximate solution based on the MLE ${\hat \theta}=t({\hat \eta})$ using asymptotic variance ${\hat \sigma}$ is not very satisfactory, particularly if t is nonlinear. The author first develops theory of signed distances $X\sb{\theta}$ on the surface $t(\eta)=\theta$ and shows that $X\sp 2\sb{\theta}$ is connected with the LR statistic. He proposes bootstrap confidence intervals, and shows that these are invariant under transformations of y and $\theta$ which are continuously differentiable one-one mappings of the corresponding spaces into itself. The bootstrap confidence intervals thus automatically produce accurate confidence intervals for problems which can be transformed to multivariate normal with known variance-covariance matrix.
Reviewer: B.K.Kale

MSC:
62F25Parametric tolerance and confidence regions
62F12Asymptotic properties of parametric estimators
62H12Multivariate estimation
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