At the boundary null

π_{1} = π_{0} = π

\overset{π}{ˉ} = π

. Independence of the binomial proportions gives

E T = 0, Var T = 1,

asymptotically, so

T \Rightarrow N (0, 1)

. For the composite null

δ < 0

, the mean moves to the nonrejection side, making the boundary the least favorable case.

ii

Solution

Under

δ = δ_{A}

E T ≃ \frac{δ _{A}}{π ˉ ( 1 - π ˉ ) ( 1/ n _{1} + 1/ n _{0} )},

and

Var T ≃ \frac{π _{1} ( 1 - π _{1} ) / n _{1} + π _{0} ( 1 - π _{0} ) / n _{0}}{π ˉ ( 1 - π ˉ ) ( 1/ n _{1} + 1/ n _{0} )} .

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