Obtaining a PhD title in statistics without new theoretical results (i.e. theorems)?

Question

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.

Related questions:

1. What happens if a math PhD student fails to find a proof that is the main objective of his/her thesis topic?
2. PhD theses in mathematics with no "big" results or no results at all

Answer 1

PhDs are assessed in different ways in different places, and even in the same place there is considerable variation between different reviewers. So a general answer can't be given.

That said, as a statistician I surely think that a PhD-worthy thesis can be produced without proving original theorems. A number of statistics PhD theses are applied; they may involve modelling and new methodological developments; these may not be groundbreaking but rather adapt things that already exist to the specific data to be analysed. Theory is not always (easily) available for methodological novelties, and quite a bit of methodological research is done via systematic simulations. A well done simulation study, a new methodological idea, and a competent discussion of existing literature can carry major weight of a PhD thesis. On top of that, in applied theses there can be issues such as handling missing values, and making sense of a large body of non-statistical literature that has implications on the analysis.

I have seen certain examples where PhD candidates do "theory for the sake of having some theory", and I didn't find this very convincing, even though it cannot denied that in this way a certain ability is demonstrated that some people expect. (These people probably are in Mathematics departments; elsewhere I don't think this is generally expected.)

Note also that some mathematical theory is rather easily available, for example showing whether an ad-hoc distance is actually a metric, i.e., fulfills the triangle inequality (it depends on the distance whether that's easy of course). I have seen a number of people introducing maximum likelihood estimators for some model, and one could show that these are asymptotically normal by just checking the assumptions of standard theory.

As a thesis reviewer I would criticise lack of theory if the thesis, in my view, "calls out" for such theory (e.g., asymptotic normality of the ML estimator is used but not proved, and neither in the literature for the specific instance), and I convince myself that the theory would have been easy enough to do. (If it's not there, I'd like to see a discussion and reasons why not.) But this only concerns specific theses, and would in most cases on its own not be a reason for failure anyway.

Note also that your situation, doing a statistics PhD without a very strong mathematical background, and lack of training and confidence with proving theorems, is not unusual. Many others have been in the same situation, and I believe that most of them eventually got their PhD.