I have arena mirrors and they were installed in accordance with the glass company's instructions, using a plywood backing and glass adhesive.  That works well, but it's really difficult to replace a mirror, as I discovered when my Mum hit one with a spur and broke it.

Since installing ours, I've come across a much better way of handling installation.  Cover the back of the mirrors with two layers of MacTac, which stabilizes the mirror and prevents it from falling into the floor in pieces if it breaks.  The only frame necessary is a wood channel for the bottom and the top, and that can be as simple as a length of 2"x4" across the bottom and the top, and secured on both ends.  Bear in mind that the mirrors usually need some angling into the arena to be truly useful, and that they should be installed above the kicking boards.  I've also had very good luck with top quality mirrored closet doors, which already have backing and frame, so they're easier to handle, and can be easily installed into the channel frame.

I live in a relatively damp climate, with temperature extremes (-35 to +40), and I have not had any difficulties with the mirrors in the seven years they've been up, but they are installed in my indoor, so they are protected from direct sun and rain and snow.