The majority of digital ballasts have internal protection circuitry designed to cut power in the event of overheating or short circuit. Since it actually powers on for some time before shutting off I'd lean towards overtemp protection shutting it off, typically with a short it would show up immediately upon power on except in some odd edge cases.
If I was going to attempt to troubleshoot it, I'd start by pulling the case off & checking for any discolored components(particularly the markings on the components change color from white to brownish). After that, measure resistance across the capacitors, you should see either an open circuit or some resistance, a 0 ohm measurement indicates a problem. Finally I'd plug it in & measure voltage rails as well as checking for components that get excessively hot. Do NOT attempt this if you are not comfortable working with electronics, plugging it in with the case off can expose you to LETHAL voltages. Hell, be careful even with it not plugged in because capacitors can hold dangerous amounts of energy if they don't have any bleed-off resistors.
Unfortunately though most problems with digital ballasts will be difficult to troubleshoot without a schematic unless it's something obviously blown, and even then you may not find the problem that caused it to blow in the first place. And like queequeg said, a lot of them are encased in resin to prevent you working on them even if you can determine what's wrong.
I'm an electronics tech by trade & honestly I'd either see if I could RMA it or just scrap it if I could afford to replace it.
