I am a student of the Statistics Department. And now I can choose a subject from such list: numerical mathematics (analysis), differential equations, complex analysis, real analysis. Can you please tell me how can we use this courses in statistics and are they important for statistics at all? And which course should I choose?
I think the best thing for you to do is to talk to your academic advisor if you haven't done so already. The reason I say that is my guess that an actual professor will have more information on:
- the requirement and customs of this specific department
- general availability of the classes you mentioned and frequency with which they are offered
- the order in which students from your cohort usually take classes
- you personal preferences and proclivities
That is assuming you have an advisor who is relatively easily accessible and who will be willing to spend time on a brief conversation with you. This is the best case scenario, as in addition to aforementioned advantages related to the access of information, talking to the advisor gives you an option for real-time dialogue with the person advising.
If, however, talking to academic advisor is not a viable option for you, I would recommend taking the courses you listed in the following order:
- real analysis
- complex analysis/differential equations
- numerical analysis
I assume you already have solid background in (multivariable) calculus and linear algebra. If not, you should take these to ASAP, and definitely before you take anything from your list.
Also, I want to point out again that whatever I wrote should be taken with a grain of salt, as it is intended to be a "rule of thumb" list. In order to make proper decision on which classes to take when, you should take into account a lot of individual-specific factors, included but not limited to
- you background
- academic interests
- customs of university and department
- who is teaching which class any given semester
- which topics are covered in each of the courses
- whether it is possible (and acceptable for you) to take several classes simultaneously
I would guess the department policy behind requiring one of these courses is to promote a broader perspective on the theory of statistics, and to develop "mathematical maturity".
Often statisticians are called upon to do things with data that, from a purely logical point of view, are impossible. It would be helpful to me in this situation to have knowledge that lets me know my limits, if not necessarily to forcefully argue with end-users what those limitations are.
Of the topics listed, numerical methods and real analysis are the most closely related to statistics because they encompass the very important practice and theory of approximation. A sizable amount of statistics involves assessing the quality of fitting models to data, so knowing the theory behind this activity would be useful. For example, it gives one a better understanding of "degrees of freedom" in applying statistical inference methods.
That said, I would see in differential equations and complex analysis good material for a better understanding of the special functions that statisticians are constantly employing (e.g. the "error function"). While less directly applicable to statistics, these might be better choices for someone who desires a comprehensive understanding of the mathematics behind the methods of mathematical statistics.