Good Books on Intro Statistics with Proofs and Theory?

[Bipolarity]

After having read the first half of Introduction to Statistics and Data Analysis by Peck and Olsen, I must say that the book is complete garbage.

Does anyone know any good books on intro statistics that provides proofs for its theorems and the axioms of probability? It should be focused more on theory than on application. I don't mind if it uses any calculus.

Thanks!

BiP

 

[Stephen Tashi]

Does anyone know any good books on intro statistics that provides proofs for its theorems and the axioms of probability?

Axioms are assumptions, so they aren't proven. Which theorems are you talking about? What are some examples?

 

[Bipolarity]

Axioms are assumptions, so they aren't proven. Which theorems are you talking about? What are some examples?

Sorry I meant "provide proof for its theorems and provide axioms of probability". Ambiguity in English language :D

For example, the Central Limit Theorem and the Law of Large Numbers.

BiP

 

[micromass]

Do you want a textbook on probability theory or on statistics. If you want proofs of the CLT and of the law or large numbers, then it's probability theory you need.

A good book is "Probability and measure" by Billingsley. It's not the easiest book though, but the proofs of the CLT and of the SLLN are not easy anyway.

 

[Stephen Tashi]

Let me expand on micromass's observation that textbooks on introductory statistics and textbooks on probability tend to be two completely different things - because it's one of my favorite soap boxes.

The type of statistics in introductory textbooks ("frequentist" statistics) is focused on quantifying the probability of observed data given certain ideas are assumed. The statistical tests that use these calculations are simply procedures. They don't prove any result. They don't even quantify the probability of a hypothesis about the data being true or false. So you can't expect to see a proof of their correctness. (Look up "optimal statistical decisons" if you want to see the context in which statistical tests can be proven optimal.)

The proofs relevant to frequentist statistics are those that establish the correctness of calculations for the probability of the data, and various functions of the data.

To do probability theory in an absolutely rigorous manner requires measure theory ( -at least that's the current state of mathematical knowledge). Bilingsley's book uses measure theory. However, measure theory is extremely abstract and it's actually hard to relate measure theory to practical problems in probabiity or statistics.

 

[camillio]

In addition to above posts recommending to think of what you really need - if it's probability, I double Billingsley. Feller's Intro to probability is also worth a look, as well as Kolmogorovs' Foundations of probability and Loeve's Probability theory. They are nice but hard texts.

Regarding statistics (not Bayesian), I propose Kendall's Advanced theory of statistics, which is a little old but still nice text. Shao's Mathematical statistics, I find nice, but sometimes people grumble about it.

Bayesian statistics - I like Robert's Bayesian choice, Bayesian data analysis by Gelman et al. Bernardo's Bayesian theory is famous, but for me it's difficult to read, maybe it lacks some rigour or whatever.

 

[chiro]

After having read the first half of Introduction to Statistics and Data Analysis by Peck and Olsen, I must say that the book is complete garbage.

Does anyone know any good books on intro statistics that provides proofs for its theorems and the axioms of probability? It should be focused more on theory than on application. I don't mind if it uses any calculus.

Thanks!

BiP

The kinds of books that get into theoretical probability using measure theory are the books that cover stochastic calculus in the most general sense.

Shreve has written about this kind of thing in both discrete and continuous time, and there are a lot of books that cover pretty much the same thing.

I wouldn't recommend you do the general treatment without getting a little intuition first though, because the intuition is what is needed for solving the more practical problems - even if they are requiring you to use all the generalized notions.

Also for martingales, there is a probability called Probability With Martingales published under Cambridge University Press. You should read this as well if you want to expand your probability theory understanding.

For statistics, this is dependent on what type of statistical analysis you want to learn, but at a minimum I would get a detailed book on linear models of all kinds as a start.

 

[DrDu]

Personally I like Cox and Hinkley, Theoretical Statistics. It provides a thorough basis on all kinds of tests and statistics with both illuminating examples and mathematical proofs (without nitpicking).

 

[20Tauri]

This is the book we used in my mathematical statistics class. We didn't go through the whole thing, and I think it leans more towards probability than statistics, but it does involve a fair amount of proof. It's pretty clear and straightforward. Of course, I took the class with the professor who wrote it, so I am a bit biased. It also relies a lot on Mathematica use, so that might not be helpful for you.

https://www.amazon.com/dp/1420079387/?tag=pfamazon01-20