One professor at our institute gave all his students the same advice when it comes to Ph.D. application: apply to non-obvious programs at top universities. Here "obvious programs" refer to the program that aligns well with your area of study. For example, if you are a Master's student in Statistics, then Ph.D. programs in Statistics, Applied Math, or Computer Science are obvious choices, whereas those in Applied Psychology, Electrical Engineering, and Mechanical Engineering are the non-obvious programs he suggests applying to.
His rationale is that the title of a top school helps a lot in job hunting (particularly in the industry), so you should try to get into a school with good prestige. However, obvious programs in these schools will be very competitive with strong applicants having the same background as you, so you should apply to less popular programs instead. He believes the mismatch in area of study is not going to be an issue, because Ph.D. students from every department do applied math in their research, so our quantitive skills will be useful no matter where we go.
I wonder how far should I take this advice. In fact, I do have some interest in electrical engineering, but some schools only allow one application, so I can apply to one program only. On the one hand, I always find hardware accelerators very cool and own a few FPGAs as a hobbyist. In fact, I taught myself Verilog in my gap year (it was the pandemic), which is quite unusual for someone with a Bachelor's in Statistics, so you can tell I do have some passion for EE. Additionally, I have some research experience in deep learning with JAX on TPUs, which means I am at least a user of state-of-the-art hardware accelerators. Moreover, I wrote some CUDA in a course project and am quite good at programming in general, which the recommendation letters can hopefully prove. Finally, for better or worse, my mathematical background makes me a special applicant. On the other hand, I have absolutely zero research experience in EE and few relevant courses on my transcript, as I stopped playing with electronics when I enrolled at a top research university as a Master's student in Statistics: I never consider the option of pursuing a Ph.D. in EE, because I thought it only was for those who have a relevant background.
Is it a good idea for me to apply to EE instead of Statistics? If this question is too narrow for this site, can someone possibly convince the admission committee in the EE department that he/she will be a good Ph.D. candidate with little relevant research experience?