In this paper we propose the zero-inflated COM-Poisson distribution. We develop a Bayesian analysis for our model via on Markov chain Monte Carlo methods. We discuss regression modeling and model selection, as well as, develop case deletion influence diagnostics for the joint posterior distribution based on the psi-divergence, which has several divergence measures as particular cases, such as the Kullback-Leibler (K-L), J-distance, L-1 norm and chi(2)-square divergence measures. The performance of our approach is illustrated in an artificial dataset as well as in a real dataset on an apple cultivar experiment. (C) 2014 Elsevier B.V. All rights reserved.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Sao Paulo State Univ, Fac Engn Bauru, Sao Paulo, BrazilUniv Sao Paulo, Dept Appl Maths & Stat, BR-05508 Sao Paulo, BrazilSao Paulo State Univ, Fac Engn Bauru, Sao Paulo, Brazil