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I am looking for a nonparametric statistical theory textbook that does not avoid tools from measure-theoretic probability and covers proofs on topics outside of rank-based hypothesis tests: e.g., kernel density estimation, nonparametric estimation, and resampling methods (i.e., bootstrapping, jackknife).

The closest textbook I can find to this is Tsybakov's Introduction to Nonparametric Estimation. What can I use to supplement this textbook?

Wasserman's All of Nonparametric Statistics has the appropriate topic coverage, but does not really cover proofs.

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  • $\begingroup$ Do you think Tsybakov's book is good for beginners who want an introduction that isn't as proof or measure theory heavy? I searched for it and found some small previews which make it seem good, but I couldn't tell for sure. $\endgroup$
    – roundsquare
    Aug 13, 2020 at 19:54
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    $\begingroup$ @roundsquare If you want to avoid proofs and measure theory entirely, something like Bishop's Pattern Recognition and Machine Learning would do. $\endgroup$
    – Clarinetist
    Aug 13, 2020 at 20:55
  • $\begingroup$ Thanks @Clarinetist. I took a look at the table of contents - it doesn't seem to be focused on nonparametric estimation - at least to me since I don't know much about the subject. Is that the focus of Bishop's book? $\endgroup$
    – roundsquare
    Aug 14, 2020 at 20:54
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    $\begingroup$ @roundsquare It is not, but it covers the basics avoiding measure theory and proofs. In all honesty, I haven't seen a single nonparametric estimation that avoids proofs entirely. Another book to consider might be Semiparametric Regression by Ruppert, Wand, and Carroll - but be warned, this text is quite dense. I would honestly recommend you ask a question too; you'll probably get better recommendations that way. $\endgroup$
    – Clarinetist
    Aug 14, 2020 at 20:59
  • $\begingroup$ Thank you very much for asking this question! I only saw rank-based proofs before since the nonparametrics course clashed with other coursework I had to take. After finding Györfi et al's book, I also picked up a copy. $\endgroup$
    – kurtosis
    Aug 15, 2020 at 22:04

2 Answers 2

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I would recommend A Distribution-Free Theory of Nonparametric Regression by Györfi, Kohler, Krzyżak and Walk. It looks like what you are looking for: proofs use sigma fields as needed, there is plenty of use of dominated convergence and Fatou's Lemma, and it hits topics like kernels and k-nearest neighbors.

The online notes from László Györfi seem like a smaller (i.e. much less complete) version of what you are looking for: there are sigma fields, dominated convergence proofs, etc. Maybe take a peek at that to see if you want to investigate the larger book by him and coauthors.

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There's a chapter in this book on nonparametric statistics: "Modern Statistical Methods for Astronomy With R Applications", Feigelson and Babu https://www.cambridge.org/us/academic/subjects/physics/astronomy-general/modern-statistical-methods-astronomy-r-applications?format=HB&isbn=9780521767279

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