Modeling sparse data such as microbiome and transcriptomics (RNA-seq) data is very challenging due to the exceeded number of zeros and skewness of the distribution. Many probabilistic models have been used for modeling sparse data, including Poisson, negative binomial, zero-inflated Poisson, and zero-inflated negative binomial models. One way to identify the most appropriate probabilistic models for zero-inflated or hurdle models is based on the p-value of the Kolmogorov-Smirnov (KS) test. The main challenge for identifying the probabilistic model is that the model parameters are typically unknown in practice. This paper derives the maximum likelihood estimator (MLE) for a general class of zero-inflated and hurdle models. We also derive the corresponding Fisher information matrices for exploring the estimator's asymptotic properties. We include new probabilistic models such as zero-inflated beta binomial and zero-inflated beta negative binomial models. Our application to microbiome data shows that our new models are more appropriate for modeling microbiome data than commonly used models in the literature.