Take the model X N(,1). Consider testing H0 : 2 {0,1} against H1 :
Question:
Take the model X » N(¹,1). Consider testing H0 : ¹ 2 {0,1} against H1 : ¹ Ý {0,1}. Consider the test statistic T Æ min{j p
nXnj, j p
n(Xn ¡1)j}.
Let the critical value be the 1¡® quantile of the random variable min{jZj, jZ ¡
p nj}, where Z » N(0,1).
Show that P
£
T È c j ¹ Æ 0
¤
Æ P
£
T È c j ¹ Æ 1
¤
Æ ®. Conclude that the size of the test Án Æ 1{T È c} is ®.
Hint: Use the fact that Z and ¡Z have the same distribution.
This is an example where the null distribution is the same under different points in a composite null.
The test Án Æ 1(T È
c) is called a similar test because infµ02£0 P[T È c j µ Æ µ0] Æ supµ02£0 P[T È c j µ Æ µ0].
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