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J n j 1 u 2 j 1 n n i 1 u i 2 2 j n j 1 x j x 2 68

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' j n j ' 1 U 2 j & 1 n ' n i ' 1 U i 2 & ( $ $ & $ ) 2 j n j ' 1 ( X j & ¯ X ) 2 . (68) Taking expectations and using Lemma 2 and Proposition 2 it follows now from (68) that E [ ' n j ' 1 $ U 2 j ] ' ' n j ' 1 E [ U 2 j ] & 1 n E ' n i ' 1 U i 2 & E ( $ $ & $ ) 2 ' n j ' 1 ( X j & ¯ X ) 2 ' n F 2 & F 2 & F 2 ' ( n & 2) F 2 . (69) Proof of (13): SSR ' j n j ' 1 $ U 2 j ' j n j ' 1 ( Y j & $ " & $ $ . X j ) 2 ' j n j ' 1 ( Y j & ( ¯ Y & $ $ . ¯ X ) & $ $ . X j ) 2 ' j n j ' 1 ( Y j & ¯ Y ) & $ $ .( X j & ¯ X ) 2 ' j n j ' 1 ( Y j & ¯ Y ) 2 & 2 $ $ j n j ' 1 ( Y j & ¯ Y )( X j & ¯ X ) % $ $ 2 j n j ' 1 ( X j & ¯ X ) 2 ' j n j ' 1 ( Y j & ¯ Y ) 2 & $ $ 2 j n j ' 1 ( X j & ¯ X ) 2 . (70) Proof of (28): It follows from (3) that Y n % 1 & $ Y n % 1 ' U n % 1 & ( $ " & " ) & ( $ $ & $ ). X n % 1 ' U n % 1 & j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 . U j & j n j ' 1 X n % 1 ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 U j ' U n % 1 & j n j ' 1 1 n % ( X n % 1 & ¯ X )( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 . U j . (71)
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29 Proof of (29): It follows from (28) and Lemma 3 that F 2 Y n % 1 & $ Y n % 1 ' F 2 % j n j ' 1 1 n % ( X n % 1 & ¯ X )( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 2 . F 2 ' F 2 1 % 1 n % 2 n . ( X n % 1 & ¯ X ) ' n j ' 1 ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 % ( X n % 1 & ¯ X ) 2 ' n j ' 1 ( X j & ¯ X ) 2 ( ' n i ' 1 ( X i & ¯ X ) 2 ) 2 ' F 2 n % 1 n % ( X n % 1 & ¯ X ) 2 ' n j ' 1 ( X j & ¯ X ) 2 . (72)
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