Whose nirvana is it anyway? Allow me to take you on a little detour through Heidegger?

He claims that "my death" is inalienably mine: that no one can die my death for me.

I think this means that "my death" never arrives... That may sound weird, but if we view death as a complete extinction of the "person" or whatever makes them up, then the empirical instant that death begins necessarily goes unnoticed - as in I can have no cognition of "my death" or by proxy the time it occurs.

i.e. It cannot be said that the event of my death is in any way only empircally real for me (because such a quality would never arrive): but who else dies for me? Which of course isn't very Buddhist...

But it does make me wonder if a similar thing can be said about nirvanic extinction? Or is the true self of a Buddha his or her nirvana, such that it can be said the Buddha is no more, just not that he or she reached it?
From a buddhist point of view, i think the question is flawed already from the beginning by assuming that there exists a "self or person" by using the words "whose nirvana, my death, person, his or her nirvana". 

By using these words it is implied that a self exists which it do not. There exist a physical and a mental stream and nowhere is there a self to be found. A self is not even needed in the equation. Things work just fine without the need for a self.

If you try to search for this self by either using the method of analysis where you break things up into the constituent parts and look at them you will not find a self anywhere.

If you use the method of synthesis by looking at things in a relational/interdependent way by using the doctrine of dependent origination you will also not find a self anywhere. Why? Because according to dependent origination things only exist as a part/reflection of other things. Therefore no real self can be found. 

The false idea of a self arises when the 5 aggregates work together as a physio-psychological machine. The false idea of a self is really just one of the 52 mental formations belonging to the 4th aggreate and all the aggregates are subject to the 3 signs of existence; anicca, dukkha, anatta. This is said by Dr. Walpola Rahula on p. 18 in his book "What The Buddha Taught".

So in order to answer your question i would say that you should take out the aforementioned words and then you have your answer which is that no "self or person" achieves nibbana. Assuming that a "self or person" exists is not refering to ultimate reality - paramattha sacca. Instead you are dealing with concepts which is not real.

Lanka