As a new Ph.D. student (mathematical statistics) how should I read papers for research? My question is about how to best read mathematics papers for research.

I just passed the last qualifying exam as a Ph.D. student in mathematics (studying statistics) and I'm reaching the point where I will need to do more research, which involves reading papers. I used to write, in my own words in a journal, pre-reading thoughts (why I'm reading this paper, what I already know, etc.), a summary of the paper's contents while reading, post-reading thoughts (what I think about the ideas, how they connect to others, how this could be useful, opinions, etc.), and a list of potentially interesting bibliography entries for later reading. When reading I would essentially read linearly, making margin notes, highlighting interesting sections, and so on. Reading a single paper could take a day, going through this process.

This did well enough for my undergraduate thesis on a topic in economics. It also worked well for the paper that my adviser and I are wrapping up and submitting to journals. But I get the impression that this is not an optimal way to read papers, especially in subjects as technically intense as mathematics. (My role in the paper I co-authored with my adviser was to do simulation studies for a test statistic he and another co-author developed, which is more computational and programming intensive than mathematically intensive, and paper reading was basically to get background on the subject. This is fine for me as a starting researcher, but I don't think I will get by on this alone for a Ph.D. thesis, let alone in my later career.)

I'm shy about asking my adviser the best way to read papers, so I'd like to hear others' approach. Is my approach to paper reading a good one, or is there a better way?
My PhD was in statistics, and I have also supervised PhD students in this field, so hopefully I can offer some useful advice.  Here are some thoughts:


Talk to your supervisor: There is no room for intellectual shyness in that relationship.  Your supervisor is not going to be upset with you for asking questions, evening if it is something you think you should know how to do already.  The fact that you even think to think about this is a good sign, and your process is quite impressive.  Don't be shy.
Streamline your process: Your process sounds quite comprehensive already.  Personally, I find that writing a short summary in your own words/notation is probably the most useful exercise to understand math papers.  Some of the other stuff you're doing might be a bit excessive, especially if you're only getting through one paper a day.
Learn the maths, expand your tool-kit: In a mathematical field, in additional to understanding the specifics of the paper you're reading, you also want to use this as an opportunity to gradually expand your mathematical tool-kit.  To do this, make sure you understand the derivation/proof of each result, and if you see an unfamiliar mathematical technique, talk to your supervisor about why it works, and when you might deploy it.
Learn to "see the matrix": Think about the intuition of each method and and make sure you can describe it intuitively.  When reading mathematical theorems, ask yourself lots of questions:  Do I understand this result and its relevance to the field?  How would this be applied in practice?  Do I understand the proof(s)?  Are there other ways to prove this?  If there are necessary conditions for a result, do I understand why they're necessary (i.e., why does the result fail without them)?  If there are sufficient conditions for a result, do I understand why these are sufficient (i.e., why does the result hold in this case)?
Jot down your own ideas: As you read the literature, jot down your own ideas for possible extensions and interesting questions that come out of this.  At first these might be quite ill-posed, but you can gradually refine them as you learn more.