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Improvements of interval estimations for the variance and the ratio of two variances. (English) Zbl 0715.62072
Summary: Let $$S_ 0$$ and $$S_ 1$$ be random variables distributed independently as $$\sigma^ 2\chi^ 2(n_ 0)$$ and $$\sigma^ 2\chi^ 2(n_ 1,\lambda)$$, respectively, where $$\chi^ 2(n_ 0)$$ denotes the central chi-square distribution with $$n_ 0$$ as degrees of freedom (d.f.) and $$\chi^ 2(n_ 1,\lambda)$$ denotes the noncentral chi-square distribution with $$n_ 1$$ as d.f. and noncentrality parameter $$\lambda$$. A confidence interval for $$\sigma^ 1$$ is usually constructed based only on $$S_ 0$$ and its form is $$[S_ 0/c_ 2,S_ 0/c_ 1]$$ with $$c_ 1$$ and $$c_ 2$$ some nonzero constants. A confidence interval based on both $$S_ 0$$ and $$S_ 1$$ as an improvement to this existing procedure is proposed. It is linked with the Stein’s estimator for $$\sigma^ 2$$ which dominates the best equivariant estimator. The improvement of the interval estimation for the ratio of two variances has also been discussed.

MSC:
 62F25 Parametric tolerance and confidence regions