On Two-Stage Confidence Interval Procedures and Their Comparisons for Estimating the Difference of Normal Means

Abstract

We consider two independent N(μ₁, σ²) and N(μ₂, a ²σ²) populations where μ₁, μ₂, σ² are unknown parameters with −∞ < μ₁, μ₂ < ∞, 0 < σ < ∞. We assume that a (>0) is known! The problem is one of estimating Δ = μ₁ − μ₂ by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient at least 1 − α. Here, d (>0) and 0 < α < 1 are both preassigned numbers. First, a two-stage procedure 𝒫₁ is designed in the spirit of Stein (1945, Annals of Mathematical Statistics 16: 243–258). Then, another two-stage procedure 𝒫₂ is tried in the spirit of Chapman (1950, Annals of Mathematical Statistics 21: 601–606). We report that the Stein-type two-stage procedure 𝒫₁ performs better than 𝒫₂. Next, a variant of a two-stage procedure due to Aoshima et al. (1996, Sequential Analysis 15: 61–70) is included as our third procedure, 𝒫₃. In a variety of situations, we find that 𝒫₁ also comes out ahead of 𝒫₃. A new alternative sampling design 𝒫₄ is then introduced by incorporating a two-stage sampling technique from one population alone followed by a single-stage sampling strategy from the other population. We observe that 𝒫₄ compares favorably with 𝒫₁ with regard to the achieved confidence level whereas the margin of oversampling in the case of 𝒫₄ is justifiably smaller than that associated with 𝒫₁. Additionally, 𝒫₄ has a significant operational edge over 𝒫₁ and hence we suggest implementing 𝒫₄ in practice. In the end, we illustrate superiority of the new procedure 𝒫₄ with an example using a real data set from horticulture (Mukhopadhyay et al., 2004, Journal of Agricultural, Biological and Environmental Statistics 38: 1384–1391).

Keywords:

• Behrens–Fisher problem
• Fixed-width interval
• Operational simplicity
• Oversampling
• Proportional variance
• Single-stage sampling
• Two-stage sampling

Subject Classifications:

• 62L12
• 62L05
• 62J15

ACKNOWLEDGMENTS

We thank two referees and an Associate Editor for their helpful commentaries and many constructive suggestions.

Notes

τ₃ row 1 corresponds to a = 0.1, 0.5, 0.6, 0.7 (from left to right) and τ₃ row 2 corresponds to a = 1.0, 1.1, 1.5, 2.0 (from left to right).

Recommended by Makoto Aoshima