The Structure of Bivariate Distributions

Reduced Rank Models for Contingency Tables

“…If p -1 canonical correlations are nonzero, then II can be written in the form A p , with w, and dj equal to the marginals ir i+ and TT+J, with A, equal to the canonical correlations, and with x fa and y js equal to the canonical scores (Lancaster, 1958). Thus C p implies A p .…”

Reduced rank models for contingency tables

“…We also compare R p with the model suggested by correspondence analysis, written as A p , in which Another way of formulating A p is by saying that II has a Fisher-decomposition of rank p -1 (Lancaster, 1958 Proof. If p -1 canonical correlations are nonzero, then II can be written in the form A p , with w, and dj equal to the marginals ir i+ and TT+J, with A, equal to the canonical correlations, and with x fa and y js equal to the canonical scores (Lancaster, 1958).…”

The general theory of canonical correlation and its relation to functional analysis

“…The analogy with the classical theory is now almost complete for B will be generated by the square integrable kernel where f x is orthogonal to all ^, u x to all y> t . This is the case discussed, for j> = q= 1, by Lancaster [1]. Lancaster begins from Karl Pearson's coefficient of mean square contingency, & 2 , which he generalises defining it (in our notation) by…”

“…By a well known property of bounded linear functionals on a Hilbert space we know that (1) U.«] where A is a linear operator from Jt to JV, and putting u = Af in (1) we derive, from Schwartz's inequality, that [3] The general theory of canonical correlation 231…”

“…Lancaster [1] has extended this theory, for p = q = 1, to include a description of the correlation of any function of a random variable x and any function of a random variable y (both functions having finite variance) for a class of joint distributions of x and y which is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space.…”