Confidence Intervals and Sets for Unknown Parameters in Statistical Distributions, Study notes of Mathematical Statistics

Download Confidence Intervals and Sets for Unknown Parameters in Statistical Distributions and more Study notes Mathematical Statistics in PDF only on Docsity! TA: Yuan Jiang Email: jiangy@stat.wisc.edu STAT 710: Discussion #23 April 22, 2008 1 Construction of Confidence Sets Example 1. Let X1, ..., Xn be a random sample of random variables with Lebesgue density θaθx−(θ+1)I(a,∞)(x), where θ > 0 and a > 0. (i) When θ is known, derive a confidence interval for a with confidence co- efficient 1 − α by using the cumulative distribution function of the smallest order statistic X(1). (ii) When both a and θ are unknown and n ≥ 2, derive a confidence interval for θ with confidence coefficient 1 − α by using the cumulative distribution function of T = ∏n i=1(Xi/X(1)). (iii) Show that the confidence intervals in (i) and (ii) can be obtained using pivotal quantities. (iv) When both a and θ are unknown, construct a confidence set for (a, θ) with confidence coefficient 1 − α by using a pivotal quantity. Example 2. Let X be a sample of size 1 from the negative binomial dis- tribution with a known size r and an unknown probability p ∈ (0, 1). Using the cumulative distribution function of T = X − r, show that a level 1 − α confidence interval for p is [ 1 1 + T+1 r F2(T+1),2r,α2 , r T F2r,2T,α1 1 + r T F2r,2T,α1 ] , where α1 + α2 = α, Fa,b,α is the (1−α)th quantile of the F-distribution Fa,b. Office: 1275A MSC 1 Phone: 262-1577