Mathematical Basis of Statistics: Probability and Mathematical Statistics: A Series of Monographs and Textbooks by Jean-René Barra (PDF)

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Text: English, French (translation)

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⭐This is not a handbook of statistics! It is a mathematics text that develops the full depth of the theory.It is typical in math grad school to have a semester long course in real analysis (measure theory, Lebesgue integration, convergence theorems, Fourier analysis, L^p spaces, and some stuff on differentiation). After that you’re ready for a proper course in probability and random processes (probability spaces as measure spaces, martingales, all the assorted bounds, large deviation theorems).I ran across this book while looking for a statistics book to follow up from here. Barra begins by defining a statistical space as a sigma algebra with some family of probability measures on it, and follows this up by defining a dominated statistical space as one where there is a single measure in terms of which we can define all the probability measures of integrals of some functions (the densities). He spends three chapters setting up this careful apparatus (including the definition of Fisher information) before moving on to hypothesis testing.My only argument with this book is his lack of counterexamples. The exercises seem reasonable, but he does not, for instance, give an example of a statistical space which is not dominated.

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