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stat210a_fall07_hw2

stat210a_fall07_hw2 - UC Berkeley Department of Statistics...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 2 Fall 2007 Issued: Thursday, September 6 Due: Thursday, September 13 Problem 2.1 Suppose that ( X 1 , X 2 , . . . , X n ) is multivariate normal with unknown mean ( μ, . . . , μ ), where each component has the same mean μ R and covariance matrix σ 2 I . First, suppose that σ 2 > 0 is known. Show that T = n i =1 X i is sufficient for the parameterization θ = μ in two ways: (a) using the factorization criterion, and (b) by direct use of the definition of sufficiency. Now suppose that σ 2 > 0 is not known, and the model is parameterized by θ = ( μ, σ 2 ). Find and establish sufficiency of a non-trivial set of sufficient statistics (i.e., not the data itself). Problem 2.2 Consider a family P = { p ( x ; θ ) | θ Ω } , where p ( x ; θ ) are density functions with common support X . (a) Prove that T is sufficient if and only if for each θ, θ 0 Ω, the likelihood ratio p ( x ; θ ) p ( x ; θ 0 ) depends on x only via T ( x ).
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