What does a random contingency table look like?
Abstract
Let R=(r1, ..., rm) and C=(c1, ..., cn) be positive integer vectors such that r1 +... + rm=c1 +... + cn. We consider the set Sigma(R, C) of non-negative mxn integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D in Sigma(R,C) is close with high probability to a particular matrix ("typical table'') Z defined as follows. We let g(x)=(x+1) ln(x+1)-x ln x for non-negative x and let g(X)=sumij g(xij) for a non-negative matrix X=(xij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative mxn matrices X with row sums R and column sums C at a unique point, which we call the typical table Z.
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