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).