When we take X 1 , . . . , X n from F (...

Will Osborn

Will Osborn

Answered

2022-11-24

When we take X 1 , . . . , X n from F ( x ). Then the ranks are X ( 1 ) < . . . < X ( n ) and lets take the test-statistic,
t 0 ( X 1 , . . . , X n ) = i = 1 n X ( i )
If we look at t ( X 1 , . . . , X n ) to see if t ( X 1 , . . . , X n ) = ? t ( X 1 , . . . , X n ), does the rank change?
For example is then X 1 , . . . , X n correspond to X ( 1 ) < . . . < X ( n ) ?

Answer & Explanation

Jayda King

Jayda King

Expert

2022-11-25Added 8 answers

This test statistic is simply
t ( X 1 , , X n ) = i = 1 n X ( i ) = i = 1 n X i = n X ¯ n
If you flip the signs of each summand then you get t ( X 1 , , X n ) = n X ¯ n

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