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

摘要

We consider two independent N(μ_1,σ~2) and N(μ_2, a~2σ~2) populations where μ_1,μ_2, σ~2 are unknown parameters with -∞ < μ_1,μ_2 < ∞,0 < σ < ∞. We assume that a (>0) is known! The problem is one of estimating Δ = μ_1 - μ_2 by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient at least 1 - 2. Here, d (>0) and 0 < α < 1 are both preassigned numbers. First, a two-stage procedure (P_1) is designed in the spirit of Stein (1945, Annals of Mathematical Statistics 16: 243-258). Then, another two-stage procedure (P_2) is tried in the spirit of Chapman (1950, Annals of Mathematical Statistics 21: 601-606). We report that the Stein-type two-stage procedure (P_1) performs better than (P_2) 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, (P_3) In a variety of situations, we find that (P_1) also comes out ahead of (P_3). A new alternative sampling design (P_4) 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 (P_4) compares favorably with (P_1) with regard to the achieved confidence level whereas the margin of oversampling in the case of (P_4) is justifiably smaller than that associated with (P_1). Additionally, (P_1) has a significant operational edge over (P_1) and hence we suggest implementing (P_4) in practice. In the end, we illustrate superiority of the new procedure (P_4) with an example using a real data set from horticulture (Mukhopadhyay et al., 2004, Journal of Agricultural, Biological and Environmental Statistics 38: 1384-1391).