Stats Discussion 12: Confidence, Prediction Intervals, & Likelihood Ratios - Prof. Yi Chai, Exams of Statistics

Download Stats Discussion 12: Confidence, Prediction Intervals, & Likelihood Ratios - Prof. Yi Chai and more Exams Statistics in PDF only on Docsity! STAT 610 DISCUSSION 12 TA: Yi Chai Office: 1335D MSC Email: chaiyi@stat.wisc.edu Webpage: http://www.stat.wisc.edu/∼chaiyi Office Hours: 2:00-4:00pm Wednesday or by appointment 1. Confidence bounds, intervals and regions • Example 1 (4.5.1): Let X1, · · · , Xn1 and Y1, · · · , Yn2 be independent exponential E(θ) and E(λ) samples, respectively, and let ∆ = θ/λ. (a) If f(α) denotes the αth quantile of F2n1,2n2 distribution, show that [Ȳ f( 1 2α)/X̄, Ȳ f(1 − 1 2α)/X̄] is a confidence interval for ∆ with confidence coefficient 1− α. (b) Show that the test with acceptance region [f(12α) ≤ X̄/Ȳ ≤ f(1− 1 2α)] has size α for testing H : ∆ = 1 versus K : ∆ 6= 1. • Example 2 (4.6.1): Suppose X1, · · · , Xn is a sample from a Γ(p, 1θ ) distribution, where p is known and θ is unknown. Exhibit the UMA level (1− α) UCB for θ. 2. Prediction intervals • Example 3 (4.8.2): Let X1, · · · , Xn+1 be i.i.d. as X ∼ F , where X1, · · · , Xn are observable and Xn+1 is to be predicted. A level (1− α) lower(upper) prediction bound on Y = Xn+1 is defined to be a function Y (Y ) of X1, · · · , Xn such that P (Y ≤ Y ) ≥ 1− α (P (Y ≤ Y ) ≥ 1− α). (a) If F is N(µ, σ20) with σ 2 0 known, give level (1−α) lower and upper prediction bound for Xn+1. (b) If F is N(µ, σ2) with σ2 unknown, give level (1 − α) lower and upper prediction bound for Xn+1. (c) If F is continuous with a positive density f on (a, b), −∞ ≤ a < b ≤ ∞, give level (1 − α) distribution free lower and upper prediction bounds for Xn+1. 3. Likelihood Ratio Procedures • Example 4 (4.9.9): The normally distributed random variables X1, · · · , Xn are said to be serially correlated or to follow an autoregressive model if we can write Xi = θXi−1 + i, i = 1, · · · , n, where X0 = 0 and 1, · · · , n are independent N(0, σ2) random variables. (a) Show that the density of X = (X1, · · · , Xn) is p(x, θ) = (2πσ2)− 1 2 n exp{− 1 2σ2 n∑ i=1 (xi − θxi−1)2} for −∞ < xi < ∞, i = 1, · · · , n, x0 = 0. (b) Show that the likelihood ratio statistic of H : θ = 0 (independence) versus K : θ 6= 0 (serial correlation) is equivalent to −( ∑n i=2 XiXi−1) 2/ ∑n i=1(X 2 i ). 1