Take the model X N(,1). Consider testing H0 : 2 {0,1} against H1 : {0,1}. Consider the test statistic T

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].