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 ($\in \mathcal{R}$) when the population variance ${\sigma }^2$ ($\in {\mathcal{R}}^{+}$) also remains unknown. With stopping variables constructed based on an arbitrary general estimator W_m 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 W_m, 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.

摘要

在本文中，我们设计了一种创新的通用类改进的两阶段抽样方案，以增强双抽样和改进的双抽样程序。在平方误差损失加上抽样的线性成本下，我们重新审视了未知正态均值μ的最小风险点估计（MRPE）的经典问题（$\in \mathcal{R}$) 当总体方差${\sigma }^{\mathrm{2个}}$($\in {\mathcal{R}}^{+}$) 也仍然未知。停止变量基于满足一组特定条件的任意一般估计量 W m 构造，我们_的程序被证明具有渐近一阶和二阶效率以及渐近一阶风险效率。为了便于说明，我们进一步研究了具体的修改后的两阶段 MRPE 程序，其中我们用样本标准差的适当倍数、基尼平均差(GMD) 和平均绝对差(MAD) 代替W _m， 分别。广泛的模拟研究被用来验证我们的理论发现。然后分析女性厌食症患者体重变化的真实数据集，以证明这些改进的两阶段 MRPE 程序的实际适用性。在试验样本中存在可疑异常值的情况下比较它们，我们凭经验相信基于 GMD 和 MAD 的程序看起来比基于样本标准差的程序更稳健。