Imperial Fame is what you get from assault (and possibly some various Al Zahbi quests) that allows you to rank up.
You do not need to do all 5 missions of that rank to rank up. You can repeatedly do one mission of the current rank you are and get Superior Private having only done 2 assaults.
You get a certain amount of "points" per successful assault. These "points" are what I call
Imperial Fame.
The only way you can lose Imperial Fame that I know of is failing the quest Finding Faults and I know this because RDM of my static did same amount of Assault I did and he got the quest after recieving his Yigit turban, I did not. Other ways of losing
Imperial Fame probably include losing so many consecutive assaults.