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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.

62F25 Parametric tolerance and confidence regions