Medians

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...