2 Questions for you lot. One I don't really know and one that I'm struggling to unravel my thoughts with...

First one that I'm unsure of...

Let \phi (x) be a strictly increasing function of a real variable. Show that the result of applying the sign test to a sample X_1, \dots , X_n with hypothetical median M_0 is identical to that of applying the test to the transformed sample \phi (X_1), \dots , \phi(X_n) with hypothetical median \phi (M_0).

And the other one
Consider a random sample X_1, \dots ,X_n from an unknown distribution F(x - \theta) where \theta is unknown and F(0) = \frac{1}{2}, i.e. \theta is the population median. Show that the Hodges-Lehmann point estimate of \theta arising from the Sign Test is the sample median of the data.

Now in the notes it says that when \theta = M, \theta is symmetric around some a_0 which occurs when a_0 = \frac{n}{2}.

So its symmetrical around X_{\frac{n}{2}} which if you performed the sign test on would show its the median...! Hellary I cant really explain my thoughts I have no idea how I'm gonna write this down...