Count data with excess zeros and ones arise frequently in many practical application fields such as traffic safety, public health, risk management and so on. In this paper, in order to study this kind of data set more deeply, a zero-and-one-inflated geometric distribution model is proposed and considered. Jeffreys prior and reference priors are derived via data augmentation strategy and the complete likelihood function. For different sample sizes and different true values of the parameters, numerical simulation and comparative analysis are adopted to assess the performance of these different objective priors. Finally, one accidental data set from Detroit is analyzed to illustrate the practicability of the proposed model. It shows that the objective Bayesian method performs better than the subjective Bayesian method.