Let X 1 , X 2 , . . . , X n be a...

Aleah Avery

Aleah Avery

Answered

2022-11-23

Let X 1 , X 2 , . . . , X n be a random sample from a N ( θ 1 , θ 2 ) distribution. Find the uniformly minimum variance unbiased estimator of 3 θ 2 2 .

Answer & Explanation

Kristen Garza

Kristen Garza

Expert

2022-11-24Added 13 answers

Lets X i N ( μ , σ 2 ). We want to show ( X i , X i 2 ) is complete for ( μ , σ 2 ).
It is enough to show ( X ¯ , S = ( X i X ¯ ) 2 ) is complete. We know X ¯ and S are independent and X ¯ N ( μ , σ 2 n ), S G a m m a ( n 1 2 , 2 σ 2 ).
We should show if ( μ , σ 2 )
E ( g ( X ¯ , S ) ) = 0 P ( g ( X ¯ , S ) = 0 ) = 1
0 = E ( g ( X ¯ , S ) ) = 0 + g ( x ¯ , s ) f ( x ¯ ) f ( s ) d x ¯ d s
= 1 Γ ( n 1 2 ) ( σ 2 ) n 1 2 0 ( + g ( x ¯ , s ) f ( x ¯ ) s n 1 2 1 e s σ 2 d x ¯ ) d s
= 1 Γ ( n 1 2 ) ( σ 2 ) n 1 2 0 ( + g ( x ¯ , s ) f ( x ¯ ) s n 1 2 1 d x ¯ ) e s σ 2 d s
= 1 Γ ( n 1 2 ) ( σ 2 ) n 1 2 0 ( h ( s ) ) e s σ 2 d s
The above is a Laplace transform of h ( s ), which implies h ( s ) = 0, a.e.
So
0 = + g ( x ¯ , s ) f ( x ¯ ) d x ¯
= + g ( x ¯ , s ) 1 2 π σ 2 n e n 2 σ 2 ( x ¯ μ ) 2 d x ¯
= + ( g ( x ¯ , s ) 1 2 π σ 2 n e n 2 σ 2 x ¯ 2 e n 2 σ 2 μ 2 ) e n 2 σ 2 2 x ¯ μ d x ¯
The above is a Two-sided Laplace transform.
So g ( x ¯ , s ) = 0 a.e.

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