{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Dr. Hackney STA Solutions pg 89

# Dr. Hackney STA Solutions pg 89 - Second Edition 6-3 e Fix...

This preview shows page 1. Sign up to view the full content.

Second Edition 6-3 e. Fix sample points x and y . Define A ( θ ) = { i : x i θ } , B ( θ ) = { i : y i θ } , a ( θ ) = the number of elements in A ( θ ) and b ( θ ) = the number of elements in B ( θ ). Then the function f ( x | θ ) /f ( y | θ ) depends on θ only through the function n i =1 | x i - θ | - n i =1 | y i - θ | = i A ( θ ) ( θ - x i ) + i A ( θ ) c ( x i - θ ) - i B ( θ ) ( θ - y i ) - i B ( θ ) c ( y i - θ ) = ( a ( θ ) - [ n - a ( θ )] - b ( θ ) + [ n - b ( θ )]) θ + - i A ( θ ) x i + i A ( θ ) c x i + i B ( θ ) y i - i B ( θ ) c y i = 2( a ( θ ) - b ( θ )) θ + - i A ( θ ) x i + i A ( θ ) c x i + i B ( θ ) y i - i B ( θ ) c y i . Consider an interval of θ s that does not contain any x i s or y i s. The second term is constant on such an interval. The first term will be constant, on the interval if and only if a ( θ ) = b ( θ ). This will be true for all such intervals if and only if the order statistics for x are the same as the order statistics for y . Therefore, the order statistics are a minimal sufficient statistic.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online