Uniform association in contingency tables associated to Csiszár divergence
Consider an \(I\times J\) contingency table corresponding to positive probabilities \((p_{ij})\). Define \[ r_{ij}= (p_{ij}p_{i+1,j+1})/(p_{i+1,j}p_{i,j+1}),\quad r=\sum_{i=1}^{I-1}\sum_{j=1}^{J-1} r_{ij},\quad \pi_{ij}=r_{ij}/r, \] for \(i=1,\dots,I-1\), \(j=1,\dots,J-1\), and \(\pi_u=1/[(I-1)(J-1)]\). Let \({\mathbf {\pi}}=(\pi_{11}, \dots, \pi_{I-1,J-1})\) and \({\mathbf {\pi}}_u=(\pi_u, \dots,\pi_u)\). The uniform association in the contingency table means that \({\mathbf {\pi}}={\mathbf {\pi}}_u\). If \(\phi\) is a convex function then the Csiszár divergence is \[ D_{\phi}({\mathbf {\pi}}_u,{\mathbf {\pi}})= \sum_{i=1}^{I-1}\sum_{j=1}^{J-1}\pi_{ij}\phi(\pi_u/\pi_{ij}). \] The authors propose a measure of uniform association \[ U_{\phi}({\mathbf \pi}_u,{\mathbf {\pi}})= [D_{\phi} ({\mathbf {\pi}}_u, {\mathbf {\pi}})-\phi(1)]/[\phi(\pi_u)-\phi(1)] \] and derive the asymptotic distribution of its sample version. The results are used for constructing statistical tests.
scientific article; zbMATH DE number 1124490