hw3_stat210a_solutions

hw3_stat210a_solutions - UC Berkeley Department of...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3 Fall 2006 Issued: Thursday, September 14, 2006 Due: Thursday, September 21, 2006 Graded exercises Problem 3.1 (a) To prove that, we massage the density expression into its exponential family form: f ( x ; λ,μ ) = bracketleftbigg- λ 2 1 x- μ 2 x + radicalbig λμ + 1 2 log parenleftbigg λ 2 π parenrightbiggbracketrightbigg x − 3 / 2 I ( x > 0) = [ η ( λ,μ ) · T ( x ) + B ( λ,μ )] h ( x ) with: T ( x ) = bracketleftbig 1 x x bracketrightbig η ( λ,μ ) = bracketleftbig- λ 2- μ 2 bracketrightbig h ( x ) = x − 3 2 I ( x > 0) B ( λ,μ ) = 1 2 log parenleftbigg λ 2 π parenrightbigg + radicalbig λμ (b) To show that this is a scale family, we consider Y = σX (with σ > 0) and show that its density can be expressed in the IG format for suitably chosen ˜ λ and ˜ μ . We know that: f Y ( y ) = 1 σ f X ( y σ ) Therefore: f Y ( y ) = 1 σ bracketleftbigg- λ 2 σ y- μ 2 y σ + radicalbig λμ + 1 2 log parenleftbigg λ 2 π parenrightbiggbracketrightbigg parenleftBig y σ parenrightBig − 3 / 2 I ( y σ > 0) = bracketleftbigg- λσ 2 1 y- μ/σ 2 y + radicalbigg ( σλ ) parenleftBig μ σ parenrightBig + 1 2 log parenleftbigg σλ 2 π parenrightbiggbracketrightbigg y − 3 / 2 I ( y > 0) which proves that Y has IG distribution with parameters ( ˜ λ, ˜ μ ) = ( σλ, μ σ ) (c) n productdisplay i =1 f ( x i ; λ,μ ) = bracketleftBigg- λ 2 n summationdisplay i =1 1 x i- μ 2 n summationdisplay i =1 x i + radicalbig λμ + 1 2 log parenleftbigg λ 2 π parenrightbigg bracketrightBigg x − 3 / 2 I ( x > 0) 1 Since there exist no a and b such that a ∑ n i =1 X i + b ∑ n i =1 X − 1 i = 0 almost surely and the interior of the parameter space R × R is nonempty, this is a full rank exponential family and hence T ( X ) = ( ∑ i X i , ∑ i X − 1 i ) is a complete sufficient statistic. Now notice let ˜ T ( X ) = parenleftBig ¯ X, ∑ i parenleftBig 1 X i- 1 ¯ X parenrightBigparenrightBig and notice that ˜ T 1 ( X ) = T 1 ( X ) n and ˜ T 2 ( X ) = T 2 ( X )- n 2 T 1 ( X ) which is a one-to-one transform of T ( X ) implying that ˜ T ( X ) is also a complete statistic. Problem 3.2 To determine the natural parameter space of each of the families, we must determine the set A T,h = { η : integraltext h ( x ) exp [ ηT ( x )] dx < ∞} . (a) In this case A h,T = R . To prove that, notice that: integraldisplay exp bracketleftbig ηx- x 2 bracketrightbig dx = exp parenleftbigg η 2 4 parenrightbiggintegraldisplay exp bracketleftbigg- parenleftBig η 2 + x parenrightBig 2 bracketrightbigg dx = √ 2 exp parenleftbigg η 2 4 parenrightbiggintegraldisplay exp bracketleftbigg- y 2 2 bracketrightbigg dy = 2 √ π exp parenleftbigg η 2 4 parenrightbiggintegraldisplay exp 1 √ 2 π bracketleftbigg- y 2 2 bracketrightbigg dy = 2 √ π exp parenleftbigg η 2 4 parenrightbigg < ∞...
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hw3_stat210a_solutions - UC Berkeley Department of...

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