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Unformatted text preview: θ , since by WLLN & 1 1 X ² + = → P , and g ( x ) = x x1 is continuous at & 1 1 + . Show that & ~ is asymptotically normally distributed ( as n → ∞ ) . Find the parameters. E ( X 1 2 ) = & ± ± ± ² ³ ´ ´ ´ µ ¶⋅ ⋅ 1 & & 1 2 & 1 dx x x = & 2 1 1 + . σ 2 = Var ( X 1 ) = ( )( ) 2 2 & & & 1 2 1 + + . By CLT, ( ) ( ) 2 ± ² , X N n D →. Since g ( x ) = x x1 is differentiable at & 1 1 ² + = , g ' ( μ ) = – ( 1 + θ ) 2 ≠ 0, by Theorem 4.3.9, ( ) ( ) ( ) ( ) ( ) ( )( ) & & ± ² ³ ³ ´ µ + + +→⋅ 2 2 2 2 & & & & N 1 2 1 1 X ± , D g g n . ¶ ( ) ( ) ( ) & & ± ² ³ ³ ´ µ + + →& & & N & & ~ 2 2 2 1 1 , D n . ¶ ( ) ( ) & & ± ² ³ ³ ´ µ + + & & & & N & ~ 2 2 2 1 1 ~ n , ( as n → ∞ )....
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Monrad
 Central Limit Theorem, Variance

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