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Stat GU4204: Final Name (Print): Spring 2020 Student UNI: Time Limit: 24 hours (take home exam) Signature: This exam contains 8 problems. Answer all of them. Point values are in parentheses. You must show your work to get credit for your solutions — correct answers without work will not be awarded points. This is an open book, open notes exam. However, you cannot collaborate/discuss with your class- mates. 1 10 pts 2 15 pts 3 10 pts 4 10 pts 5 15 pts 6 15 pts 7 10 pts 8 15 pts TOTAL 100 pts Stat GU4204 Spring 2020 Final - Page 2 of 14 May 16, 2020 1. (10 points) (6 + 4) A random sample X1, . . . , Xn is drawn from a population with p.d.f. fθ(x) = 1 2 (1 + θx), x ∈ [−1, 1], and fθ(x) = 0 if x /∈ [−1, 1], where θ ∈ [−1, 1] is the unknown parameter. Find an unbiased estimator of θ and show that it is consistent. Stat GU4204 Spring 2020 Final - Page 3 of 14 May 16, 2020 2. (15 points) (5 + 5 + 5) Suppose that we have two independent random samples: X1, . . . , Xn iid∼ Exp(θ) (thus, E(X1) = θ−1), and Y1, . . . , Ym iid∼ Exp(µ), where θ, µ > 0 are unknown. (a) Find the likelihood ratio test (LRT) for H0 : θ = µ versus H1 : θ 6= µ. (b) Show that the LRT in part (a) can be based on the statistic T = ∑n i=1Xi∑n i=1Xi+ ∑m j=1 Yj . Stat GU4204 Spring 2020 Final - Page 4 of 14 May 16, 2020 (c) Find the distribution of T when H0 is true. Stat GU4204 Spring 2020 Final - Page 5 of 14 May 16, 2020 3. (10 points) (4 + 4 + 2) Suppose that we have a random sampleX1, X2, . . . , Xn from Uniform([0, θ]), where θ > 0 is unknown. Suppose that we want to test the following hypothesis: H0 : 3 ≤ θ ≤ 4, versus H1 : θ < 3="" or="" θ=""> 4. (1) Let Yn = max{X1, . . . , Xn}. Consider the following two tests: δ1 : Reject H0 if Yn ≤ 2.9 or Yn ≥ 4 and δ2 : Reject H0 if Yn ≤ 2.9 or Yn ≥ 4.5. (a) Find the power functions of δ1 and δ2, when θ ≤ 4. Stat GU4204 Spring 2020 Final - Page 6 of 14 May 16, 2020 (b) Find the power functions of δ1 and δ2, when θ > 4. (c) Which of the two tests seems better for testing the hypothesis (1)? Stat GU4204 Spring 2020 Final - Page 7 of 14 May 16, 2020 4. (10 points) (5+5) Consider two particles situated at locations X1 and X2 on the horizontal axis and particles Y1 and Y2 located on the vertical axis. It may be assumed that all the random variables are independent. Furthermore, X1 and X2 are i.i.d. N(0, σ 2 1) and Y1 and Y2 are i.i.d. N(0, σ22). You only observe X1−X2 and Y1−Y2 based on which you want to estimate the ratio of standard deviations σ1/σ2. (i) Show that H(X1, X2, Y1, Y2, σ1, σ2) = (X1 −X2)2σ22 (Y1 − Y2)2σ21 is a pivot. What is its distribution? (ii) If σ1 = σ2 = σ, calculate the distribution of (X1−X2)2+(Y1−Y2)2 2σ2 and indicate how your can construct a confidence interval for σ2 based on the above expression. Stat GU4204 Spring 2020 Final - Page 8 of 14 May 16, 2020 5. (15 points) (3 + 4 + 3 + 5) Suppose that X1, X2, . . . , Xn are i.i.d N(θ, 1), where θ ∈ R is unknown. Let ψ = Pθ(X1 > 0). (a) Find the maximum likelihood estimator ψ̂ of ψ. (b) Find an approximate 95% confidence interval for ψ. Stat GU4204 Spring 2020 Final - Page 9 of 14 May 16, 2020 (c) Let Yi = 1{Xi > 0}, for i = 1, . . . , n. Define ψ̃ = (1/n) ∑n i=1 Yi. Show that ψ̃ is a consistent estimator of ψ. (d) Which estimator of ψ, ψ̂ or ψ̃, is more preferable in this model. Stat GU4204 Spring 2020 Final - Page 10 of 14 May 16, 2020 6. (15 points) (3 + 3 + 4 + 5) We obtain observations Y1, . . . , Yn which can be described by the relationship Yi = θx 2 i + �i, where x1, . . . , xn are fixed constants and �1, . . . , �n are i.i.d. N(0, σ 2) (with σ2 unknown). (i) Derive the least squares estimator θ̂ of θ. (ii) Is θ̂ unbiased? Show your steps. Stat GU4204 Spring 2020 Final - Page 11 of 14 May 16, 2020 (iii) Derive the distribution of θ̂. (iv) How would you test the hypothesis H0 : θ = 0 versus H1 : θ 6= 0? Describe the test and the critical value. Stat GU4204 Spring 2020 Final - Page 12 of 14 May 16, 2020 7. (10 points) (3 + 5 + 2) Let X1, . . . , Xn be i.i.d N(µ, σ 2), where both µ ∈ R, σ2 > 0, are unknown. Let Xn+1 be a new observation (independent of the data) from the same N(µ, σ 2) distribution. (i) Find the distribution of Xn+1 − X̄n where X̄n = ∑n i=1Xi/n. (ii) Find a statistic Tn, based on X1, . . . , Xn, such that (Xn+1 − X̄n)/Tn has a pivotal distri- bution and identify the distribution. Stat GU4204 Spring 2020 Final - Page 13 of 14 May 16, 2020 (iii) Use the above pivot to construct a (1− α) prediction interval for Xn+1? 8. (15 points) (5 + 5 + 5) Suppose that conditional on Θ = θ, X1, . . . , Xn are i.i.d Uniform(0, θ), where θ > 0. Suppose that Θ has a Gamma(n+ 1, 1/γ) distribution, i.e., π(θ) = θn γn+1Γ(n+ 1) e−θ/γ1(0,∞)(θ). (a) Find the density of X(n) = maxi=1,...,nXi, conditional on Θ = θ. Stat GU4204 Spring 2020 Final - Page 14 of 14 May 16, 2020 (b) Find the posterior density of Θ given the data, i.e., conditional density of Θ given the data. (c) Find the Bayes estimator of θ with respect to the prior density π for squared error loss.