Download Length of Confidence Intervals - Discussion #25 | STAT 710 and more Study notes Mathematical Statistics in PDF only on Docsity! TA: Yuan Jiang Email: jiangy@stat.wisc.edu STAT 710: Discussion #25 April 29, 2008 1 Lengths of Confidence Intervals Example 1. Let X = (X1, ..., Xn) (n > 1) be a random sample from the exponential distribution on the interval (θ,∞) with scale parameter θ, where θ > 0 is unknown. (i) Show that both X̄/θ and X(1)/θ are pivotal quantities, where X̄ is the sample mean and X(1) is the smallest order statistic. (ii) Obtain confidence intervals (with confidence coefficient 1−α) for θ based on the two pivotal quantities in (i). (iii) Discuss which confidence interval in (ii) is better in terms of the length. Example 2. Let θ > 0 be an unknown parameter and T > 0 be a statistic. Suppose that T/θ is a pivotal quantity having Lebesgue density f and that x2f(x) is unimodal at x0 in the sense that f(x) is nondecreasing for x ≤ x0 and f(x) is nonincreasing for x ≥ x0. Consider the following class of confidence intervals for θ: C = { [b−1T, a−1T ] : a > 0, b > 0, ∫ b a f(x)dx = 1 − α } . Show that if [b−1 ∗ T, a−1 ∗ T ] ∈ C, a2 ∗ f(a∗) = b 2 ∗ f(b∗) > 0, and a∗ ≤ x0 ≤ b∗, then the interval [b−1 ∗ T, a−1 ∗ T ] has the shortest length within C. Example 3. Let (X1, ..., Xn) be a random sample from N(µ, σ 2), µ ∈ R and σ2 > 0. (i) Suppose that µ is known. Let an and bn be constants satisfying a 2 nfn(an) = b2nfn(bn) > 0 and ∫ bn an fn(x)dx = 1 − α, where fn is the Lebesgue den- sity of the chi-square distribution χ2n. Show that the interval [b −1 n T, a −1 n T ] has the shortest length within the class of intervals of the form [b−1T, a−1T ], ∫ b a fn(x)dx = 1 − α, where T = ∑n i=1(Xi − µ)2. (ii) When µ is unknown, show that [b−1n−1(n − 1)S2, a−1n−1(n − 1)S2] has the shorest length within the class of 1 − α confidence intervals of the form [b−1(n − 1)S2, a−1(n − 1)S2], where S2 is the sample variance. (iii) Find the shortest-length interval for σ within the class of confidence Office: 1275A MSC 1 Phone: 262-1577
