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STAT 610 Midterm Exam
创建日期：2024-04-18 16:26:53
所属分类：统计学statistics
标签： STAT 610
Midterm Exam

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STAT 610 Midterm Exam 

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/ϕ/c(µ, ϕ) x > µ 0 x ≤ µ 

where θ = (µ, ϕ), ϕ > 0, µ ∈ R, and c(µ, ϕ) = ∫∞µ e−t4/ϕdt. (a) (1 point) If both µ and ϕ are unknown, is the family {fθ} 

an exponential family? Give your reasons. (b) (1 point) In the case where µ is known and ϕ is unknown, obtain a complete and

 sufficient statistic for ϕ. (c) (3 points) Suppose that ϕ = µ2 and ϕ is unknown. 

Obtain a minimal sufficient statistic. (d) (5 points) In the case where µ is unknown and ϕ = 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 posterior 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 βˆ of β is a 2-dimensional vector with first component Y¯ = 8−1 ∑8 i=1 Yi and second component 2 −1(Y¯1− Y¯2), where Y¯1 = 4−1∑4i=1 Yi and Y¯2 = 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 βˆ are independent. (c) (3 points) Show that the sum of residual squares SSR has the form SSR = 4∑ i=1 (Yi − Y¯1)2 + 8∑ i=5 (Yi − Y¯2)2 Under the condition of part (b), show by applying Basu’s theorem that SSR and Y¯ 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 βˆ is UMVUE if and only if a = b = c = d. 2 


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