post15 - Functions of a parameter Example : X 1 , , X n...

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Unformatted text preview: Functions of a parameter Example : X 1 , , X n Bernoulli( ), < < 1 Y = X i is a CSS and Y/n is UMVE of . Suppose we are interested in variance of Y/n , (1- ) n . What is the UMVE? 1 Consider ( Y/n )(1- Y/n ) n . E ( ( Y/n )(1- Y/n ) n ) = 1 n 2 EY- 1 n 3 EY 2 = 1 n (1- )- 1 n 2 (1- ) = n- 1 n (1- ) n . So n n- 1 ( Y/n )(1- Y/n ) n is an UE of (1- ) n . It is also the UMVE. In general, if T is a CSS for , then ( T ) is the unique UMVE of its expected value ( ) = E ( ( T )). 2 Case of several parameters Sufficient statistic Suppose X 1 , , X n f ( x ; 1 , 2 ). Y 1 = u 1 ( X 1 , , X n ) and Y 2 = u 2 ( X 1 , , X n ) have the joint pdf g 1 , 2 ( y 1 , y 2 ; 1 , 2 ). Then Y 1 and Y 2 are called joint sufficient statistics for 1 and 2 iff f ( x 1 ; 1 , 2 ) f ( x n ; 1 , 2 ) g 1 , 2 ( y 1 , y 2 ; 1 , 2 ) = H ( x 1 , , x n ) 3 Factorization theorem...
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post15 - Functions of a parameter Example : X 1 , , X n...

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