Recommendations for Mathematically Rigorous Texts on Statistics and Probability

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Bipolarity
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Hi PF, looking for a recommendation on a mathematically rigorous text on statistics (and perhaps probability). I have a decent amount of knowledge on linear & abstract algebra and calculus and analysis, so I don't mind if it uses them. What I don't want is a text that fails to prove its theorems with rigor. Suggestions are appreciated, thanks!

BiP
 
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I've done some looking but from what I've seen, books on statistics tend to be pricey and each looks different from the rest. So perhaps it is better to start with a book that is rigorous-looking but practical also, just to get a road map view of things.

Data Analysis - A Bayesian Tutorial might be a good way to start.

After reading this you would hopefully know where your interest lies and could guide yourself to further reading. And it would be useful to know where things fit in. Ultimately, your knowledge of statistics will be organized around how it is used. Use the "look inside" to see if this would be worthwhile.

I can't recommend a completely rigorous book because this subject is so confusing to me.

PS. Here is a cheaper book that claims to be mathematical and rigorous but I can't say if it is good or if it is what you should be learning.
 
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Since you haven't had a lot of replies, here is my two cents (from an engineer who uses staistical inference a fair amount but is *not* interested in all the rigor!)

The books by Lehmann seem to be the ones referenced all the time:
https://www.amazon.com/dp/0387988645/?tag=pfamazon01-20
https://www.amazon.com/dp/0387985026/?tag=pfamazon01-20

other books the stats people seem to like:
https://www.amazon.com/dp/0534243126/?tag=pfamazon01-20
(I own the first edition of this one, which has plenty of proofs but doesn't motivate the statistics very well to someone like me! This is the easiest of all the books I list here)

Another:
https://www.amazon.com/dp/0132306379/?tag=pfamazon01-20

Then there is the 60+ year old classic:
https://www.amazon.com/dp/0691005478/?tag=pfamazon01-20

If I were you I would go to your library and see what works for you. Don't buy any of these without looking at them first. Only YOU know what you are looking for.

best regards,

jason