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Estimating moments of a selected Pareto population under asymmetric scale invariant loss function. (English) Zbl 1393.62010

The authors study the problem of estimating moments of the selected (i.e., empirically best) out of \(k \geq 2\) Pareto populations possessing the same known shape parameter \(\beta\), but different and unknown scale parameters \(\alpha_i\), \(1 \leq i \leq k\). To this end, they assume that independent random samples of equal size \(n\) from each of the \(k\) populations are available, and they consider an asymmetric scale invariant loss function (ASIL).
First, the minimum risk equivariant estimator under ASIL, the uniformly minimum variance unbiased estimator, and the maximum likelihood estimator for \(\alpha_i^t\) are derived, where \(1 \leq i \leq k\) and \(t < \gamma = n \beta\). Three natural estimators \(\delta_E\), \(\delta_U\), and \(\delta_M\) for \(\alpha_{(k)}^t\) (the corresponding moment of the selected population) follow by applying them to the empirically best sample. The authors analyze \(\delta_E\), \(\delta_U\), and \(\delta_M\) with respect to risk-unbiasedness, consistency, and admissibility, and they compare them numerically in a simulation study.

MSC:

62F10 Point estimation
62F07 Statistical ranking and selection procedures
62C15 Admissibility in statistical decision theory
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