G4 Cube randomly shuts down/sleeps, starts up

I just got a G4 Cube from a friend that is exhibiting some very odd symptoms. The Cube is completely stock (with the exception that I replaced the PRAM battery, see below), and is the 450Mhz DVD-ROM model.

What it will do is occasionally, and for no apparent reason, will suddenly shut down. This happens completely randomly, and appears to NOT depend on what I'm doing with the computer at any particular moment. Sometimes it doesn't even make it through a complete boot cycle before it shuts down; other times, it boots up and appears to work fine, only to suddenly shut down hours (or even days) later.

At first I thought maybe it was overheating (as I have heard that the Cube is notorious for heat problems); however, this appears to not be the case. When the machine suddenly shuts down, both the Cube itself as well as the power supply do not feel noticably hot to the touch. Also, it has remained up through obviously processor intensive tasks. (For example, the other day I was playing Starcraft, and basically finished the entire Terran campaign and it didn't even bat an eyelash, but, hours later, after the machine had sat idle, I went to fire up Safari to browse a website only to have it suddenly shut down.)

Also, at least once, the machine has mysteriously STARTED UP on its own. I always shut down the machine before leaving for work in the morning, but one day I came back home to find that the machine had somehow, mysteriously STARTED UP while I was gone. Nobody was home all day, and we don't have any small animals that might have accidentally brushed up against the power switch. (no, I haven't yet called in a Ghost Hunting team to determine whether any supernatural forces were involved, although at this point I'm beginning to wonder if that may not be a bad idea...)

Anyway, like I said before, the machine was stock, except when it came to me, the PRAM battery was obviously dead (I got the "your clock is set before 1970" message on bootup). The mysterious shutdown problems started occurring right away, so I thought maybe the PRAM had gotten corrupted and was causing weird things to happen. So I obtained a replacement PRAM battery and installed it myself (yes, I observed proper static discharge precautions). While I was in there, I hit the CUDA switch for good measure. Unfortunately this didn't help.

Does anybody have any ideas as to what this might be? Flaky power supply or voltage regulator? Maybe even a bad power switch? Your help is greatly appreciated. Thanks!