Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Solutions to selected problems from a statistics course, specifically covering topics on confidence intervals, prediction intervals, and likelihood ratio procedures. Students are guided through examples using distributions such as exponential, gamma, and normal. The document also covers the concept of serially correlated random variables and the calculation of their densities and likelihood ratio statistics.

Typology: Exams

Pre 2010

1 / 1

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: [email protected] 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