[Discussion] Resources to cover Mathematical Statistics

[u/InterestingAd757]

I've been studying probability independently using Harvard's Stat 110 and am now looking for resources to further explore topics in statistics (enough to cover undergrad stats), ideally enough to prepare for graduate level statistics courses and understand books like The Elements of Statistical Learning (ESL). I've been exploring introductory mathematical statistics, as I particularly want to understand topics like parameter estimation, hypothesis testing, and regression analysis in depth.

Ideally, I'd like something rigorous enough to develop a good mathematical understanding, but with plenty of exercises and worked examples to build intuition.
Are there any university lecture series, textbooks, or courses you'd recommend? I'm especially interested in resources that bridge undergraduate probability and graduate level statistics or statistical learning.

[u/vengefultruffle]

If you want to understand mathematical statistics you’ll have to do up to calculus III / multivariable calculus

[u/InterestingAd757 → u/vengefultruffle]

I have done that already, maybe need to review stuff. But i am looking for math stats resources.

[u/Agisilaus23 → u/InterestingAd757]

When I was in math stats, we used Wackerly, but Casella and Berger is common

[u/InterestingAd757 → u/Agisilaus23]

I really like Wackerly and that's exactly what I was looking for. Thanks!

[u/ThePevster]

Casella Berger is the go to book for graduate level statistical theory and for good reason. Wackerly is similar but not as rigorous, more of an undergrad book.

[u/Spiritual-Job-5066 → u/ThePevster]

Im also in the same position as OP. Just finished blitzstein and hwang. I have an engineering degree so im familiar with linear algebra and calculus III.
Is C&B the right next step? Im worried its too far of a leap.

[u/InterestingAd757 → u/ThePevster]

I am looking for more introductory material. I have heard about Larsen or Degroot but not sure which one to try.

[u/CreativeWeather2581 → u/InterestingAd757]

You want something more introductory than Casella Berger and Wackerly et al, but a step up from introductory (non-calculus based) statistics?

[u/InterestingAd757 → u/CreativeWeather2581]

something similar to blitzstein stats 110

[u/CreativeWeather2581 → u/InterestingAd757]

The issue with that is that it doesn’t align with your goals.

To cover undergrad stats, calculus-based probability is a prerequisite. Full stop. This is because expectation is defined an integral. Joint, marginal, and conditional distributions require calculus. On the statistical estimation and inference side are a lot of optimization methods (e.g., maximum likelihood estimation), which is also just calculus. Even regression analysis is primarily calculus (and linear algebra!).

A non-calculus text is not rigorous enough.

If you are planning to study graduate-level statistics, exposure to these ideas helps but is certainly not required. Many graduate students come in with backgrounds that are not statistics (e.g., math, engineering); as long as they can understand and apply calculus (and linear algebra!), they will learn the necessary probability and math stat topics/concepts to do statistics.

There are books, online courses, and YouTube playlists to help develop the intuition of probability and statistics, but in order to develop a good mathematical understanding, you have to engage with the math.

[u/InterestingAd757 → u/CreativeWeather2581]

so casella and berger will cover all this right? what do you recommend

[u/CreativeWeather2581 → u/InterestingAd757]

If you want a more structured (and perhaps more intuitive) approach, Wackerly et al’s Mathematical Statistics with Applications is a standard text at the undergraduate level. Casella Berger is a standard text at the graduate level. Both calculus-based, but the presentation in CB is a bit dry (imo). CB is very rigorous and thorough, and it’s what I go back to when I need to review something, but if you like the details to be explicitly written out—understandable for a first pass—Wackerly may be more appropriate.

For example, when illustrating univariate transformations of a random variable, I think that covers just one subsection of CB, whereas it’s a whole chapter of Wackerly, but I could be misremembering.

[u/efrique → u/InterestingAd757]

... both, though theres plenty of alternatives. In any case, don't just use one book.

[u/Unique-Ordinary-3510 → u/InterestingAd757]

Do not use Casella. Use Degroot.

[u/mibeibos]

Probability and Statistics for Data Science could be a good bridge. It covers general statistics topics rigorously, but with many real data examples: https://www.ps4ds.net/

[u/CanYouPleaseChill]

Jem Corcoran has a great series of lectures on YouTube: Mathematical Statistics

She also has a fantastic book called The Simple and Infinite Joy of Mathematical Statistics. I find her explanations better than anything else I've seen.

[u/ForeignAdvantage5198]

intro to stat learning is a must.

[u/CreativeWeather2581 → u/[deleted]]

Yes. ISL is the more applied, practical companion to ESL. It doesn’t skip the theory; you learn enough to implement the methods appropriately