I'll be looking at this from a deathmatch perspective. Other game types is much more difficult to do this and they're not the big focus in the game (particularly with team game types).
Line of Sight, it can be a blessing or it can be a curse. Master this (or get good enough controlling this) and you will be more successful in the game.
First of all, let me point out something that could save you a lot of grief. Don't make the same mistake I did and play the game with reduced music volume. The way the game is, certain classes of sounds are controlled by the music volume. You'll notice this definitely with the videos that play. What you probably won't notice is that the pursuer whispers are also grouped with the music volume. If you play with reduced volume (I don't know if this is the case for with it completely off), you are going to have a tough time knowing if you should be playing defensively when a pursuer is around.
First things first, you need to identify your pursuer. You need to be aware at all times who your opponents are and what personas they are using. If you joined a game that started off small (4 players), you should periodically check the scoreboards so you can see how many are still in the game (people can leave or join at any time and you don't really notice the messages) and what personas to look out for.
When you hear the whispers, you should look for any player (this is where knowing which personas to look for helps) in your line of sight that remotely looks like they are trying to move closer to you, starting with the ones closest to you. Study every player in your vicinity long enough and try to figure out, where are they trying to move. You can't really do this one at a time, you have to look at everyone all at the same time. If you're a little slow or don't have the time to do this, target the nearest player around you and spam the stun button when they're close. If it was a pursuer, you will have a stun or contested kill. Otherwise if it wasn't, then nothing will happen (you cannot stun non-pursuers).
Once identified, you need to find a way for you to break their line of sight off of you while keeping them in your field of view. The best way to do that is to stand at a corner, behind a large pillar or divider so that you can hide behind while you can turn the camera to keep an eye on them.
Use your abilities if given the opportunity and it can help you. If you are able to hit them with it, try to run behind them and stun them (I find that running up in front almost always gets a contested kill if they are aware of you). You won't need to do this if you use smoke bombs or other abilities that completely incapacitates them. Otherwise, try to catch them at the corner where you're standing. Their view will most likely be from behind them and won't see you until after they round the corner. You could see them approaching and can usually get the upper-hand with your stun or in the rare cases, a guaranteed honorable death.