Mathematical statistics and data analysis 3文档.pdf

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Mathematical statistics and data analysis 3文档

LECTURE 12 Mid-term Test Date: Thursday 18 March 2010 Location : LT22 Time : 4:15 pm to 5:45 pm The mid-term test is closed book but calcu- lators and up to 3 help sheets are allowed to be brought in. 1 Case of unknown mean and un- known variance In this case, there are two parameters Θ and Ξ. A Bayesian approach requires the specification of a joint 2-dimensional prior distribution. For mathematical convenience, we assume that Θ and Ξ are independent where Θ ∼ N (θ0 , ξ−1 ), prior Ξ ∼ Γ(α, λ). Here θ0 , ξprior , α and λ are all known con- stants. 2 Then the posterior distribution of (Θ, Ξ) given the data X = (X , . . . , X ) is 1 n fΘ,Ξ |X (θ, ξ |x) ∝ fX |Θ,Ξ (x |θ, ξ)fΘ (θ)fΞ (ξ) 1 n/2 −(ξ/2) n (x −θ)2 = ξ e i=1 i C −(ξ /2)(θ−θ )2 α−1 −λξ ×e prior 0 ξ e , where C is a constant that does not depend on either θ or ξ . This posterior distribution is more compli- cated than those of the previous examples. In particular, the constant of proportional- ity C can only be evaluated using numerical integration where ∞ ∞ n/2 −(ξ/2) n (x −θ)2 C = ξ e i=1 i −∞ 0 −(ξ /2)(θ−θ )2 α−1 −λξ ×e prior 0 ξ e dξdθ.

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