STAT 610 Midterm Exam 7:00pm-8:30pm, March 15, 2021 Please show all your details/reasons for full credits. 1. Let X1, …, Xn be independent and identically distributed random variables with pdf fθ(x) = { e x 4/ /c(μ, ) x > μ 0 x ≤ μ where θ = (μ, ), > 0, μ ∈ R, and c(μ, ) = ∫∞μ e t4/ dt. (a) (1 point) If both μ and are unknown, is the family {fθ} an exponential family Give your reasons. (b) (1 point) In the case where μ is known and is unknown, obtain a complete and sufficient statistic for . (c) (3 points) Suppose that = μ2 and is unknown. Obtain a minimal sufficient statistic. (d) (5 points) In the case where μ is unknown and = 1 is known, derive a UMVUE of μ. 2. Let X1, …, Xn be independent and identically distributed positive random variables with pdf fθ(x) = θ3 2 x2e θx, x > 0, where θ > 0 is unknown. (a) (2 points) Obtain an estimator of θ using the method of moments. (b) (3 points) Derive the MLE of θ and prove that your solution is indeed the unique MLE. (c) (3 points) Obtain the UMVUE of θ. (d) (2 points) Let the prior pdf for θ to be pi(θ) = 1 Γ(α)γα θα 1e θ/γI(θ > 0), i.e., the pdf of Gamma(α, γ) with α > 0 and γ > 0. Show that the posterior pdf for θ is also a gamma pdf and calculate the posteri
or mean (the Bayes estimator) of θ. 3. Consider a linear model Y = Xβ+ε, where Y = (Y1, …, Y8) ′ is an 8-dimensional vector of observations, β = (β1, β2) ′ is a 2-dimensional vector of unknown parameters, X ′ = ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) and ε is a 8-dimensional normally distributed random vector. (a) (2 points) Show that the LSE β of β is a 2-dimensional vector with first component Yˉ = 8 1 ∑8 i=1 Yi and second component 2 1(Yˉ1 Yˉ2), where Yˉ1 = 4 1∑4i=1 Yi and Yˉ2 = 4 1∑8 i=5 Yi. (Continued on the next page) (b) (2 points) Suppose that components of ε are independent and have the same variance σ2 > 0. Show that the two components of β are independent. (c) (3 points) Show that the sum of residual squares SSR has the form SSR = 4∑ i=1 (Yi Yˉ1)2 + 8∑ i=5 (Yi Yˉ2)2 Under the condition of part (b), show by applying Basu’s theorem that SSR and Yˉ are independent. (Do not apply the Theorem on page 3 in lecture 10.) (d) (3 points) Suppose that the covariance matrix of ε is V = σ2 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a ρ ρ ρ 0 0 0 0 ρ b ρ ρ 0 0 0 0 ρ ρ c ρ 0 0 0 0 ρ ρ ρ d Show that the LSE β is UMVUE if and only if a = b = c = d. 2