Abstract:
The logarithmic series distribution is a common long-tailed distribution and has a wide range of applications in count data with positive integers, such as the species abundance in some forest and the types of fish in a sea area. In practice, however, some count data contains most of the zeros which is not suitable for logarithmic series distribution. To fit the excessive zeros in the count data, this paper extends the logarithmic series distribution to a zero-inflated logarithmic series distribution in the frame of the zero-inflated model. Three methods of parameter estimations, that are moment estimation, maximum likelihood estimation and Bayesian estimation, were used to estimate the parameters in the model. In the Bayesian estimation, the posterior distribution is constructed by the random walk metropolis algorithm since there is no analytical method for the posterior distribution. The Monte Carlo method is used to generate the simulation data of the zero-inflated logarithmic series distribution, and the mean square error is the metric which is used to compare the accuracies of different estimation methods. The results show that Bayesian method has a higher accuracy than other traditional estimation methods in case the sample size is small. Moreover, the precision of Bayesian method is comparable with the traditional method when the sample size is big, which suggests that Bayesian method has advantage in case there are only few samples. Finally, the model was used to fit the number of clinical readmissions within ninety days which has more than sixty percent zeros and led to a fairly good fitness.