Question: Let S be a statistic. Recall from class: A test is called an upper S-test if it is of the form = 1[S > a]

Let S be a statistic. Recall from class: A test is called an upper S-test if it is of the form = 1[S > a] + b1[S = a] for some real a and some 0 b 1. A test is called a lower S-test if it is of the form = 1[S < a] + b1[S = a]. for some real a and some 0 b 1.

Show the following.

(1) If is an upper S-test, then 1 is a lower S-test.

(2) If is a lower S-test, then 1 is an upper S-test.

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