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post15 - Consider(Y/n(1-Y/n n Functions of a parameter...

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Functions of a parameter Example : X 1 , · · · , X n Bernoulli( θ ), 0 < θ < 1 Y = X i is a CSS and Y/n is UMVE of θ . Suppose we are interested in variance of Y/n , θ (1 - θ ) 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 θ (1 - θ ) - 1 n 2 θ (1 - θ ) = n - 1 n θ (1 - θ ) n . So n n - 1 ( Y/n )(1 - Y/n ) n is an UE of θ (1 - θ ) n . It is also the UMVE. In general, if T is a CSS for θ , then φ ( T ) is the unique UMVE of its expected value τ ( θ ) = E ( φ ( T )). 2 Case of several parameters Sufficient statistic Suppose X 1 , · · · , X n f ( x ; θ 1 , θ 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 ; θ 1 , θ 2 ). Then Y 1 and Y 2 are called joint sufficient statistics for θ 1 and θ 2 iff f ( x 1 ; θ 1 , θ 2 ) × · · · × f ( x n ; θ 1 , θ 2 ) g 1 , 2 ( y 1 , y 2 ; θ 1 , θ 2 ) = H ( x 1 , · · · , x n ) 3 Factorization theorem The statistics Y 1 = u 1 ( X 1 , · · · , X n ) and Y 2 = u 2 ( X 1 , · · · , X n ) are called joint sufficient statis- tics for θ 1 , θ 2 iff f ( x 1 , · · · , x n ; θ 1 , θ 2 ) = k 1 ( y 1 , y 2 ; θ 1
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