A broader class of modified two-stage minimum risk point estimation procedures for a normal mean

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.

Keywords:

• Asymptotic first- and second-order properties
• Double sampling
• Gini’s mean difference
• Mean absolute deviation
• Minimum risk point estimation
• Suspect outliers

MATHEMATICS SUBJECT CLASSIFICATION:

• 62L10
• 62L12

Acknowledgements

We express our gratitude to two anonymous reviewers, the Associate Editor, and the Editor in Chief for their comments that helped improve this article.