Estimating moments of a selected Pareto population under asymmetric scale invariant loss function
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.
scientific article; zbMATH DE number 6852095
scientific article