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Which Schools to Apply To

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I am looking for honest advice about graduate programs that I might have a chance of getting accepted to.

I am an upcoming senior majoring in pure math at a small mid-tier private institution. I nearly have a 4.0 GPA (except for an A- in philosophy) and will be taking both the general GRE and math GRE in October. I have taken a variety of proof-based math classes, two comp sci classes, and will be taking probability and statistics soon. I've participated in an REU at my home university and am currently doing research with a different professor at my home university (both these projects were for pure math). I recently submitted a paper (not yet accepted) to a respected journal. I have a strong relationship with many of my professors and I will hopefully have a few good letters of recommendation. I am planning to apply for three fellowships, including the NSF-GRFP. Looking into PhD programs in Statistics, I am unsure if I would be a competitive applicant given that my school is not well-known. I have talked to a Statistics professor from my school who suggested that taking more stats classes would help my application. 

Because I am pessimistic about getting into a top-tier program for Statistics, I am wondering if anyone has any advice about what schools I could apply to. I don't want to aim too low and prevent myself from getting into a good school. But I am afraid that I might not get into any school at all if I aim too high. 

Honest feedback and responses are welcome. Thank you! :)

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