Page 479 - Probability and Statistical Inference

456 9. Confidence Interval Estimation

9.3 Two-Sample Problems

Tests of hypotheses for the equality of means or variances of two indepen-
dent populations need machineries which are different in flavor from the theory
of MP or UMP tests developed in Chapter 8. These topics are delegated to
Chapter 11. So, in order to construct confidence intervals for the difference
of means (or location parameters) and the ratio of variances (or the scale
parameters), we avoid approaching the problems through the inversion of test
procedures. Instead, we focus only on the pivotal techniques.
Here, the basic principles remain same as in Section 9.2.2. For an un-
known real valued parametric function κ(θ), suppose that we have a point
estimator which is based on a (minimal) sufficient statistic T for θ. Of-
ten, the exact distribution of will become free from θ for all
θ ∈ Θ with some τ(> 0).
If τ happens to be known, then we obtain two suitable numbers a and b
such that P [a < { – κ(θ)}/τ < b] = 1 – α. This will lead to a (1 – α) two-
θ
sided confidence interval for κ(θ).
If τ happens to be unknown, then we may estimate it by and use the
new pivot instead. If the exact distribution of { –
become free from θ for all θ ∈ Θ, then again we obtain two suitable numbers
a and b such that Once more this
will lead to a (1 – α) two-sided confidence interval for κ(θ). This is the way
we will handle the location parameter cases.
In a scale parameter case, we may proceed with a pivot whose
distribution will often remain the same for all θ ∈ Θ. Then, we obtain two
suitable numbers a and b such that . This
will lead to a (1 – α) two–sided confidence interval for κ(θ).

9.3.1 Comparing the Location Parameters

Examples include estimation of the difference of (i) the means of two inde-
pendent normal populations, (ii) the location parameters of two independent
negative exponential populations, and (iii) the means of a bivariate normal
population.
Example 9.3.1 Difference of Normal Means with a Common Un-
known Variance: Recall the Example 4.5.2 as needed. Suppose that the
2
random variables X , ..., X , are iid N(µ σ ), i = 1, 2, and that the X ’s
i1
in
i,
1j
are independent of the X ’s. We assume that all three parameters are un-
2j
+
known and θ = (µ , µ , σ) ∈ ℜ × ℜ × ℜ . With fixed α ∈ (0, 1), we wish
1
2
to construct a (1 – α) two–sided confidence interval for µ – µ (= κ(θ))
2
1