stat610hw2

# stat610hw2 - independent of the minimal suﬃcient...

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Statistics 610: Homework 2 (1) Keener’s notes Chapter 3: 11, 15, 19, 20, 31, 32. Chapter 4: 17, 19, 22, 23, 25, 30. (2) Consider X 1 ,X 2 ,...,X n i.i.d. f a,b ( x ) = (1 /b ) e - ( x - a ) /b 1( x > a ). This is called the E ( a,b ) family of distributions, with a R ,b > 0. (a)Show that ( X (1) , n i =1 ( X i - X (1) )) are minimal suﬃcient when both a,b are unknown. (b)Show also that X (1) is independent of n i =1 ( X i - X (1) ) and determine the distributions of these two statistics. (c) Show that the ratios Z i = ( X ( n ) - X ( i ) ) / ( X ( n ) - X ( n - 1) ) ,i = 1 , 2 ,..., ( n - 2) are
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Unformatted text preview: independent of the minimal suﬃcient statistic in (a). (3) (a) Let ( X 1 ,X 2 ,...,X n ) be i.i.d. U ( θ-1 / 2 ,θ + 1 / 2). Show that ( X (1) ,X ( n ) ) is minimal suﬃcient but not complete. (Hint: What happens to the distribution of X ( n )-X (1) ?) (b) Consider ( X 1 ,X 2 ,...,X n ) i.i.d. N ( θ,θ 2 ). Show that ( ∑ n i =1 X i , ∑ n i =1 ( X i-X ) 2 ) is minimal suﬃcient but not complete. 1...
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