How to test suspected hardware on Android

I have a problem with my HTC Sensation that I think is either due to problems with the electronics that interfaces to the battery or a memory problem. Basically the phone turns off spontaneously by itself every now and then. Very often - as far as I can see - when the CPU is running 100% (and hence there are a big battery drain)... Turning on the phone again can easily take 2-3 turns before it is running stable (until next time). It seems to be starting first time though if connected to power... If this occurred on a Windows PC, I would first download a hardware test/diagnostics application and see whether this could find anything... ...but for all the usual security reasons, the Android platform is way more closed than the PC/Windows platform and I suspect that it is not so easy to install a diagnostics app to test the hardware. So... how do you go about this then? Short of using a full-blown hardware analyser, which I don't have access to :-) [Note that I have rooted my device and don't mind flashing a new ROM if needed...]