Welcome to Just Answer,
My Name is John,
I fixed a 55 inch version of that same model about 2 months ago i found resister R720 220-2w overheating when set was powerd up then it would shutdown found Transisters F06-F07(C4632-ecg2588) Transisters were leaky causeing resister to overheat and go into shutdown in 1-2-sec All green LED's lit up then went out.
It might be a good idea to check E994 its a 2amp pico, and check c707 next to Horz Output Transistor.
You will need the following tools if you do not have them then this will be a very difficult repair that could cost you more to fix yourself then by a service center.
*Digital Multimeter (with diode check) (cost $ 70.00)
*Adjustable Soldering Station (this is a must a bad iron or to hot and you will damage the traces) (cost $40.00)
* Solder Supplies, Fine Solder with lead, and solder wick (cost $20.00)
You will also need the parts that can be purchased from http://www.encompassparts.com/ http://www.mcminone.com/ or even http://www.partstore.com/
If you need the service manual you can order that from http://www.service-manual.net/
Most common cause is a power surge or brown out but if the set is over 7 year old it may just be break down of the parts.
Your best bet is to invest in a good power protector, not a just a surge protector. A surge protector works great when the power company drops a power line or you get a good lighting strike. But it does nothing on a brown out which can be just as bad or worse.
Your best bet in these situations is to have a good battery backup system (they run around $50)
My Favorite so far has been the belken PureAV more pricey but worth it. You don't have to buy a specialized on for video equipment just make sure it says it does at least 1500 Volt Amps.
If you have an questions please ask.