final_2011 - Statistics 709 Final Instructor Dr Yazhen Wang...

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Statistics 709, Final Instructor: Dr. Yazhen Wang, December 20, 2011 First Name: Last Name.: Be sure to show all relevant work ! Notations : P −→ stands for convergence in probability, L p −→ for convergence in L p , d −→ for conver- gence in distribution, W −→ for weak convergence, denote by E P the (conditional or unconditional) expectation under probability measure P . 1. Suppose that X 1 , · · · , X n are i.i.d. random sample from a population with uniform distribution on [ θ, 2 θ ], θ (0 , ). (a) (3 points) Show that T n = ( X (1) , X ( n ) ) is a minimal sufficient statistics for θ . Prove or disprove that T n is complete. (b) (3 points) Show that δ n = ( X 1 + · · · + X n ) / (1 . 5 n ), ˆ θ n = X (1) , and ˜ θ n = X ( n ) / 2 are all consistent estimators of θ . (c) (3 points) Show that n ( ˆ θ n 1) d −→ G 1 and n (1 ˜ θ n ) d −→ G 2 , where G 1 and G 2 are non-degenerate distributions. Specify distributions G 1 and G 2 . (d) (1 points) Evaluate asymptotic relative efficiency of ˆ θ n w.r.t. ˜ θ n and asymptotic relative efficiency of δ n w.r.t. ˜ θ n 2. Suppose thatX1,· · ·, Xnare i.i.d.random sample and have Lebesgue pdff(x) with finitenon-zero variance. Letθ=E[|X1X2|]. (a) (4 points) Find a U-statistic estimatorUnofθand proveUnis the UMVUE ofθ.(b) (4 points) Show that asn→ ∞,n(Unθ)d−→N(0, τ2), and find an expression ofτ2 in terms of f ( x ). 3. Consider a linear model X = + ε, where ε = ( ε 1 , · · · , ε n ) has mean zero and invertible covariance matrix V . (a) (2 points) Suppose Z = 1 1 1 1 1 1 1 1 1 1 2 2 . Obtain the form of all R 3 such that τ β is estimable. 1
(b) (5 points) Suppose that Z is not of full rank, and V = σ 2 M , where σ is an unknown positive constant, and M is a known invertible matrix. Let ˜ β = ( Z τ M 1 Z ) Z τ M 1 X, τ = c τ M 1 / 2 Z, where c R n . Show E [( τ ˜ β τ β ) 2 ] = σ 2 τ ( Z τ M 1 Z ) ℓ.

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