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The problem of regions. (English) Zbl 0954.62031

Suppose that the observation space \(R^k\) is partitioned into \(m\) regions \({\mathcal R}_1\),…, \({\mathcal R}_m\). The observation of a normal vector \(y\sim N(\mu,S)\) lies in \({\mathcal R}_i\) (say). How confident should we be that it’s mean lies in \({\mathcal R}_i\)? The authors discuss bootstrap algorithms which estimate this “confidence”. The confidence is considered in the two meanings: frequentist and Bayesian. In the frequentist approach (for \(m=2\)) this confidence is defined as the attained confidence level of the test which tests the hypothesis \(H_0\): \(\mu\not\in {\mathcal R}_i\). In the Bayesian approach it is the aposteriori probability of \(\mu_i\in{\mathcal R}_i\).
There are principal difficulties in both approaches. In the frequentist setting there is a problem of choosing appropriate tests (the authors use the likelihood ratio test). In the Bayesian setting the analogous problem is to choose the prior distribution of \(\mu\). The authors equate these problems using the Welch-Peers prior in the Bayesian setting which matches frequentist confidence levels.
It is shown that the naive bootstrap “confidence” estimator is not consistent with this approach (it corresponds to the Bayesian approach with the flat prior). Improvements for the naive estimator are considered with second and third order of accuracy by \(S=S_0/n\) as \(n\to\infty\). They are based on the general \(BC_a\) and \(ABC\) bootstrap bias-correction methods. Generalizations of them on exponential families are also considered.
These results are applied to the problem of polynomial regression order estimation for cholesterol level dependence on a dose of drug and the analysis of multimodality by data on some Mexico stamps thickness.

MSC:

62F40 Bootstrap, jackknife and other resampling methods
62F25 Parametric tolerance and confidence regions
62F15 Bayesian inference
62G09 Nonparametric statistical resampling methods

Software:

bootstrap
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References:

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