Repairing these sets is a major pain - as Mitsubishi apparently Enginnered them to fail
You will have a hard time even finding someone in your area to repair.
Manufacturers typically only support products 8-10 years - and while the actual BOARDS of the TV are no longer available - they still can be repaired/refurbished by a few companies.
You're looking at a $80 service call - just to get a tech to your home. A $60-$100 diagnostic - then the repair and labor of the actual problem.
In my opinion - not a good investment of about $500+
From your description - it seems as the power supply is the actual culprit. Over time - capacitors can overheat/dry up/leak etc - until eventually the set will no longer stay powered on, as it cannot sustain the required voltages for the rest of the TV.
You'll get better picture quality, energy efficiency, and newer technology out of that $500 in a comparable LED or LCD
I'm sorry to say it would be in your best interest to invest the repair $$ in a new set.
If you have additional - as always, Just Ask.