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Assume the math background is not an issue. Which books would you recommend as the most mathematically rigorous to learn mathematical statistics at the graduate level?

Does Kendall's Advanced Theory of Statistics (3 volumes) fit the bill? Or is Schervish's Theory of Statistics better in this aspect? Or some other book?

Doesn't need to be in one book. I'm just doing this as a challenge of some sort. I have time.

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    $\begingroup$ stats.stackexchange.com/… $\endgroup$
    – Sycorax ♦
    Commented Nov 25, 2019 at 16:39
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    $\begingroup$ There was a long history in major statistical journals of sharp reviews of successive editions of the Kendall series (latterly revised by Stuart, Ord and Arnold, but of those four only Keith Ord survives and the series appears stalled) as being less than rigorous. It has other virtues, but rigour is not its main claim to attention. Wasserman's books are a nice compromise between clarity and rigour. $\endgroup$
    – Nick Cox
    Commented Nov 25, 2019 at 20:32

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The core of mathematical statistics is point estimation and hypothesis testing. From a frequentist perspective, my votes go to Theory of Point Estimation by Lehmann and Casella, and Testing Statistical Hypotheses by Lehmann and Romano.

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What area of Statistics do you want to learn? For example, if you are willing to consider Machine Learning as a branch of statistics, then Murphy (2012): Machine learning: a probabilistic perspective is a good option.

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  • $\begingroup$ I'm looking at all the standard stuff in a graduate course on mathematical statistics: distributions, inference, regression, multivariate stuff etc. Doesn't need to be in one book. I'm just doing this as a challenge of some sort. I have time. $\endgroup$
    – stranger
    Commented Nov 25, 2019 at 16:59
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    $\begingroup$ Murphy is a good book, but I don't know that I would call it "the most mathematically rigorous" or being about "mathematical statistics." A much more mathematical ML book would be e.g. Foundations of Machine Learning by Mohri, Rostamizadeh, and Talwalkar. $\endgroup$
    – Danica
    Commented Nov 25, 2019 at 17:00
  • $\begingroup$ @Dougal thank you for the comment, I myself learned something new $\endgroup$
    – PsychometStats
    Commented Nov 25, 2019 at 17:06

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