ABSTRACT
In this chapter, we pay attention to two-sample normal problems. Consider two independent sequences Xi1, . . . ,Xini , . . . of i.i.d. random variables distributed as N(µi, σ
2 i ), i = 1, 2.We assume that all four parameters
µ1, µ2, σ1, σ2 are unknown, −∞ < µ1, µ2 <∞ and 0 < σ1, σ2 <∞. Section 13.2 introduces fixed-width confidence interval problems to com-
pare the two population means, µ1 and µ2. We will construct fixed-width confidence intervals for the parameter δ ≡ µ1 − µ2 in two separate cases. We will include scenario (i) σ21 = σ
2 2 = σ
2, but the common variance σ2 is unknown, and then scenario (ii) σ21 6= σ22 . In Section 13.3, we include minimum risk point estimation problems for
comparing µ1 and µ2. We will construct minimum risk point estimators for the parameter δ ≡ µ1 − µ2 in two separate cases. We will include scenario (i) σ21 = σ
2 2 = σ
2, but the common variance σ2 is unknown, and then scenario (ii) σ21 6= σ22 . In the literature, the scenario (ii) discussed under either framework is
commonly referred to as the Behrens-Fisher problem or the Behrens-Fisher analog. We would write θ = (µ1, µ2, σ
2) in case (i) and θ = (µ1, µ2, σ 2 1 , σ
in case (ii). Sometimes, we may continue to write Pθ(.) or Eθ(.) under either scenario without explicitly mentioning what θ is. The parameter vector θ should be clear from the context.
