Is it possible to defend a Ph.D. thesis in statistics without proving any theorems? I think the answer is affirmative since I've seen a couple of theses in that fashion. However, I'm unaware of how common/rare they are.
My follow-up question is: If I doubt (and I have good reasons to do so) my abilities to prove any original results going on towards my last Ph.D. year, should I:
i) notify my advisor about it, or ii) just go through the motions (perhaps attempting to prove results as my advisor instructs me) and focus my effort on simulation studies and real-world applications.
Under many circumstances, if the advisor in question is somewhat empathetic, i) would be the better answer, as they would possibly provide alternative theoretical problems to be solved or focus their effort on simulation studies and applications. However, I'm afraid my advisor might simply give up supervising me (which again, I have good reasons to believe). That makes option ii) more tempting. The risk with this approach is that they could simply not allow me to defend a thesis without any theoretical results.
Now I will pre-emptively answer some questions that might be asked:
Why do you doubt your abilities to prove original results?: First of all, my background is not in statistics, so I feel that my "intuition" with regard to theorems in probability and statistics is behind my peers. Moreover, I haven't been able to prove even the simplest original results (conjectures, rather) that have been pointed out to me.
Top statistical journals tend to publish papers with theoretical results, is that not a worry of yours?: I plan on getting an industry job after my Ph.D., so I'm not worrying about publishing in top journals.
Why not give up on your PhD title?: That would mean having to pay back tens of thousands of dollars that I received, and I don't have that money.