Multinomial deviance ($G^2=2{\sum }_{i:y_i>0}{y}_i\log [y_i/(N{p}_i)]$)

For counts with a multinomial distribution, this is twice the log likelihood ratio of the saturated cell-probability estimate to fixed probabilities $p_i$. Zero-count terms have limiting value zero. Simulation under the specified probabilities gives a direct reference distribution when asymptotic approximations are unsuitable.