A broader class of modified two-stage minimum risk point estimation procedures for a normal mean. Issue 12 (1st December 2022) - Hu, Jun; Zhuang, Yan
Abstract: In this paper, we design an innovative and general class of modified two-stage sampling schemes to enhance double sampling and modified double sampling procedures. Under the squared error loss plus linear cost of sampling, we revisit the classic problem of minimum risk point estimation (MRPE) for an unknown normal mean μ ( ∈ R ) when the population variance σ 2 ( ∈ R + ) also remains unknown. With stopping variables constructed based on an arbitrary general estimator Wm for σ, which satisfies a set of certain conditions, our procedures are proved to enjoy asymptotic first- and second-order efficiency as well as asymptotic first-order risk efficiency. For illustrative purposes, we further investigate specific modified two-stage MRPE procedures, where we substitute appropriate multiples of sample standard deviation, Gini's mean difference (GMD), and mean absolute deviation (MAD) in the place of Wm, respectively. Extensive simulation studies are utilized to validate our theoretical findings. A real-life data set of weight change from female anorexic patients is then analyzed to demonstrate the practical applicability of these modified two-stage MRPE procedures. Comparing them in the case where there exist suspect outliers in the pilot sample, we are empirically confident that the GMD- and MAD-based procedures appear more robust than the sample-standard-deviation-based procedures.