Description
1. Exercises 6.4, 6.9, and 6.10.
2. Suppose X1, . . . , Xn are i.i.d. samples from a normal distribution N(µ, σ2
), n ≥ 2.
Prove that Pn
i=1(Xi − X¯)
2/σ2
follows a χ
2
(n − 1) distribution and it is independent
with the sample mean X¯.
3. In Example 6.4, to construct a (1 − α) × 100% confidence interval for the variance
parameter σ
2
, we assume that the lower bound is 0 and the upper bound corresponds
to a quantity involving the α-quantile of a χ
2 distribution, we now consider using α/2
and (1 − α/2)-quantiles of the same χ
2 distribution to construct another confidence
interval. It certainly will excludes 0.
(1) Give the explicit form of the new confidence interval and justify its validity by
showing the theoretical confidence level is 1 − α.
(2) Repeat the experiments in Example 6.5 with the same parameter set-up. Compare the two types of confidence interval, such as empirical coverage probability
and average confidence interval width.
(3) Repeat the experiments in Example 6.6 with the same parameter set-up. Compare the two types of confidence interval, such as empirical coverage probability
and average confidence width.
(4) Which confidence interval would you recommend in practice? Explain why.
1