The prediction risk estimation in nonlinear regression models including artificial neural networks is especially important for problems with limited data since it can be used as a tool for finding the optimal model (or network architecture) minimizing the expected risk. In this paper, we suggest the prediction risk bounds of nonlinear regression models. The suggested bounds are derived from the modulus of continuity for a multivariate function. We also present the model selection criteria referred to as the modulus of continuity information criteria (MCIC) derived from the suggested prediction risk bounds. Through the simulation for function approximation, we have shown that the suggested MCIC is effective in nonlinear model selection problems with limited data.