Asymptotic Enumeration of Binary Contingency Tables and Comparison with Independence Heuristic

Abstract

For parameters $n,\delta,B,C$, we obtained a sharp asymptotic formula for the number of $(n+\lfloor n^\delta\rfloor)^2$-dimensional binary contingency tables with non-uniform margins taking values of $\lfloor BCn\rfloor$ and $\lfloor Cn\rfloor$. Furthermore, we compared our sharp asymptotics with the classical independence heuristic estimate and proved that the independence heuristic overestimates by a factor of $e^{\Theta(n^{2\delta})}$. Our comparison is based on the analysis of the correlation ratio and an explicit bound for the constant in $\Theta$ is also obtained.