Given a sample X_1,...,X_n∼N(θ,θ^2) show, using the definition of completeness, that the statistic T=(∑_(i) X_(i),∑_(i) X^2_i) is not complete for n≥2. Use the fact that E_θ[2(∑_(i) X_(i))^2−(n+1)∑_(i) X^2_i]=0

Damien Horton 2022-07-23 Answered
Given a sample X 1 , . . . , X n N ( θ , θ 2 ) show, using the definition of completeness, that the statistic T = ( i X i , i X i 2 ) is not complete for n 2. Use the fact that E θ [ 2 ( i X i ) 2 ( n + 1 ) i X i 2 ] = 0
The statistic T ( X ) is said to be complete for the distribution of X if, for every misurable function g, E θ [ g ( T ) ] = 0 θ P θ ( g ( T ) = 0 ) = 1 θ
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Answers (1)

ab8s1k28q
Answered 2022-07-24 Author has 17 answers
Let T 1 = i X i and T 2 = i X i 2 and suppose P ( 2 T 1 2 ( n + 1 ) T 2 = 0 ) = 1. Then T 2 = 2 n + 1 T 1 2 a.s ( P ).
However by the triangle inequality,
T 1 2 T 2 = 2 n + 1 T 1 2 a . s ( P ) ,
which is a contradiction for all n > 2.
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