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The Cusp Catastrophe Model provides a promising approach for health and behavioral researchers to investigate both continuous and quantum changes in one modeling framework. However, application of the model is hindered by unresolved issues around a statistical model fitting to the data. This paper reports our exploratory work in developing a new approach to statistical cusp catastrophe modeling. In this new approach, the Cusp Catastrophe Model is cast into a statistical nonlinear regression for parameter estimation. The algorithms of the delayed convention and Maxwell convention are applied to obtain parameter estimates using maximum likelihood estimation. Through a series of simulation studies, we demonstrate that (a) parameter estimation of this statistical cusp model is unbiased, and (b) use of a bootstrapping procedure enables efficient statistical inference. To test the utility of this new method, we analyze survey data collected for an NIH-funded project providing HIV-prevention education to adolescents in the Bahamas. We found that the results can be more reasonably explained by our approach than other existing methods. Additional research is needed to establish this new approach as the most reliable method for fitting the cusp catastrophe model. Further research should focus on additional theoretical analysis, extension of the model for analyzing categorical and counting data, and additional applications in analyzing different data types.