709-11-exam3 - Statistics 709 Exam 3 Instructor Dr Yazhen...

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Statistics 709, Exam 3 Instructor: Dr. Yazhen Wang, November 23, 2011 First Name: Last Name.: Be sure to show all relevant formulas and work ! Notations : P -→ stands for convergence in probability, denote by E Q the (conditional or uncondi- tional) expectation under probability measure Q . 1. Suppose that ( X 1 , Y 1 ) , · · · , ( X n , Y n ) are i.i.d random sample from a population P η with a joint pdf f η ( x, y ) = η 2 exp( - ηx - ηy )1( x > 0 , y > 0) , η (0 , ) . Let θ = P η ( X 1 + Y 1 1). a) (1 points) Show that 1( X 1 + Y 1 1) is an unbiased estimator of θ . b) (3 points) Find the UMVUE of θ . 2. Suppose that X 1 , · · · , X n are a i.i.d. random sample from a population with uniform distri- bution on [ - θ, θ ], θ (0 , ). a) (5 points) Show that S n = max( - X (1) , X ( n ) ) is a minimal sufficient statistics for θ . Prove or disprove that S n is complete. c) (3 points) Show that T n = ( | X (1) | + | X ( n ) | ) / 2 is a consistent estimator of θ . 3. A decision problem has a parametric family of continuous distributions P θ , θ (0 , 1), i.i.d random sample X = ( X 1 , · · · , X n ) ( n > 1) with X i (0 , 1), loss function L ( · , · ). Suppose that T is a sufficient statistics for θ , S n ( X ) is a non-randomized decision rule, and δ ( x , A ) is

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