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I need a graduate level, mathematically rigorous textbook on multivariate analysis and inference to enhance myself. I've been reading the Elements of Statistical Learning and doing the problems within, but I need a book with other focuses. Topics like famous distributions (Wishart, Wilks lambda, etc.), hypothesis testing, some theory on estimation (point, interval), and other more modern materials are welcome. I've checked this question, but the OP there was looking for something useful for his psychological analysis, which is not exactly what I want here. Currently I have the An Introduction to Multivariate Statistical Analysis, so I would also like to hear the comments about this book and its exercises. Thank you.

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Anderson is probably the most mathematical of the existing textbooks, and as such is orthogonal to the material in HTF. So this could be a good fit for you. I wrote an Amazon list on multivariate books when I was looking for one to teach both an elective doctorate course, and an applied master's course. Mardia, Kent and Bibby and Rencher top that list, so either one could be a decent alternative if Anderson proves to be too heavy.

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My graduate program at UCSB taught with An Introduction to Multivariate Analysis and it is a very good book for learning the basic tools however, it is not overly theoretical or filled with too many proofs (I always felt like it was an undergrad text). However, I do think its an excellent book in solidifying the concepts and the exercises are is very good at actually allowing you to use the methods/concepts as if you were applying them to the real world and not just deriving theory. Just my two cents.

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On the other hand, if you want something still more theoretical than Anderson's book, you can go with Muirhead: "Aspects of Multivariate Statistical Theory". This is if you really want to enter the theory of the Wishart distribution. A book that focuses on the mathematics behind multivariate statistics is Farrell: "Multivariate calculation: use of the continuous groups" which I also have found useful.

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