Is there a test that shows starter is going out

Special equipment is usually required to test the Stater Motor. The fact that the vehicle cranks over and starts with a proper jump...one would assume the starter is good and you have a bad connection at the replacement battery and or starter motor. The diagnosis would start with a Voltmeter across the battery...and doing voltage drop test though out the Cranking Circuit. If a proper jump start continues to work....then.....you need to go back and check the battery and Alt. Once again ...that Volt Ohm Amp Meter is needed for proper diagnosis.