Page 567 - Probability and Statistical Inference

544 12. Large-Sample Inference

In Chapter 9 we had a similar confidence interval for the unknown mean with
exact confidence coefficient 1 − α. That methodology worked for all sample
sizes, large or small, but at the expense of the assumption that the population
distribution was normal. The present confidence interval from (12.3.3) works
for large n, whatever be the population distribution as long as s is positive and
finite. We do realize that the associated confidence coefficient is claimed only
approximately 1 − α.
Next, let us briefly discuss the two-sample problems. Suppose that the
random variables X , ..., X are iid from the i population having a common
th
i1
ini
pmf or pdf f (x ; θ ) where the parameter vector θ is unknown or the func-
i i i i
tional form of f itself is unknown, i = 1, 2. Let us assume that the variance
i
th
of the i population is finite and positive which in turn implies

that its mean µ ≡ µ (θ ) is also finite, i = 1, 2. Assume that µ σ are unknown,
2
i
i
i
i
i
i = 1, 2. Let us also suppose that the X ’s are independent of the X ’s and
1j 2j
denote the sample mean and sample variance
2. In this case, one can
invoke the following form of the CLT: If n → ∞, n → 8 such that n /n → d
1
2
2
1
for some 0 < δ < ∞, then
Thus, using Slutsky’s Theorem, we can immediately conclude: If n → ∞, n 2
1
→ ∞ such that n /n → δ for some 0 < δ, < ∞, then
1 2
For large sample sizes n and n , we should be able to use the random variable
1 2
as an approximate pivot
since its asymptotic distribution is N(0, 1) which is free from µ , µ , σ and
2
1
1
σ . With preassigned α ∈ (0, 1), we claim that
2

which leads to the confidence interval