The first thing you would want to check for in the set, are the capacitors visually, making sure none are bulged, swollen or leaking fluid. The Power Supply is the board inside the TV that power cord directly attaches to.
Any that do not appear cylindrical, or are leaking you would take the rating (In uf), and the voltage off the cap (example: 2200 uf, 50 vdc), and cross reference them at the sites below:
They each solder into place with two leads, a positive and a negative. Make sure you solder the news ones in the same way, or they will explode when the TV is powered up.
Make sure you are careful not to un-solder any other adjacent circuit traces when removing the old caps, and carefully solder in the new ones allowing the solder to flow into the joint. Replace any and ALL caps that are questionable. Remember- when in doubt, change them out.
Try powering up the TV, and if the set operates normally, then you have solved the issue by replacing the faulty cap(s).
If it does not, then you would need to isolate which board is causing the problem by measuring the Power Supply Voltages.
Please only proceed if you have experience measuring live voltages on a circuit board level.
Use a DVM (Digital Volt Meter) set to DC volts, put the Black probe of the Meter on an RCA jack shield on the back of the TV, then use the Red probe and probe all of the Outputs for voltage (correct voltages will be stamped on the power supply at the outputs). Do this with NOTHING connected to the Power supply but AC power, the Mainboard, and the little board that houses the Power Switch. If a voltage is missing/ fluctuating, then the power board needs to be replaced.
If the power supply outputs its rated voltages, then connect the other boards one at a time until one causes it to shut down. This board would have the short causing the supply to SOS, and would need to be replaced.
The Mainboard is located at the link below:
The Power Supply is at the following link:
Between both boards you're looking at around $150 with shipping.