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Causal Inference in Statistics: A Primer, JudeaPearl, MadelynGlymour and Nicholas P.Jewell, John Wiley & Sons, 2019, 156 pages, $46.75, paperback ISBN: 978‐1‐1191‐8684‐7

Year: 2020
Type: book-review
Abstract: A k-circulant A k;n (1 = k = n-1) is an n-square matrix whose each row is obtained from the previous by k circular shifts to the right.Its first row (x 0 , . . ., x n 1 ) is called the input.Nothing is said about the x i 's in the definitions, but I guess that they are real numbers.The matrix A k;n is a circulant C n if k = 1 and a reverse circulant RC n if k = n -1.(The term 'k-circulant' has also another meaning.It is often defined that a matrix is a k-circulant if it is obtained from C n by multiplying with k its all entries below the main diagonal.)If the sequence (x 1 , . . ., x n 1 ) is palindromic, then C n is symmetric, denoted by SC n .Also, RC n is symmetric.The empirical spectral distribution function (ESD) of an n-square matrix is obtained by putting a mass 1=n at its each eigenvalue.The limiting spectral distribution function (LSD) of a sequence of n-square matrices is the weak limit (if it exists) of the sequence of their ESDs.In the case of random matrices, this limit is understood in some probabilistic sense.A description of some main points of Chapters 1-8 and 11 follows.The titles of Chapters 9 and 10 are informative enough.Chapter 1.The matrices A k;n , C n , RC n , and SC n are defined.Call them 'circulant-type matrices'.A formula for the eigenvalues of A k;n is given.This formula is fundamental in what follows (except Chapter 2).Chapter 2. The ESD and LSD are defined.A general technique to find the LSD of symmetric random matrices, based on the moment method, is introduced and applied to RC n and SC n when the input is i.i.d.(i.e. it consists of independent and identically distributed random variables).
DOI: 10.1111/insr.12369
Language: English
Open Access status: green
OpenAlex ID: W3043933609
Cites: 4
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