Usually, the performance of control charts are widely measured by average run length (ARL). In this paper, the derivative explicit formulas of the ARL for double moving average (DMA) control chart are proposed for monitoring the process mean of zero-inflated Poisson integer-valued autoregressive first-order (ZIPINAR(1)) model. This model is fit when there are an excessive number of zeros in the count data. The performance of the DMA control chart is compared with the results of moving average and Shewhart control charts by considering from out of control average run length (ARL1). The numerical results found that the DMA control chart performed better than other control charts in order to detect mean shift in the process. In addition, the real-world application of the DMA control chart for ZIPINAR(1) process is addressed.

We introduce the new estimator of odds ratios in rare events using Empirical Bayes method in two independent binomial distributions. We compare the proposed estimates of odds ratios with two estimators, modified maximum likelihood estimator (MMLE) and modified median unbiased estimator (MMUE), using the Estimated Relative Error (ERE) as a criterion of comparison. It is found that the new estimator is more efficient when compared to the other methods.

The odds ratio estimation when observed frequencies are very small usually causes difficulty in calculation. In this paper, we proposed the estimator of odds ratio for small count using Empirical Bayes (EB) in Zero-inflated Poisson distribution, where the hyper-parameters can be estimated via the posterior marginal distribution function. We compare the proposed estimator of odds ratio based on EB in Zero-inflated Poisson distribution with moments method estimator (MME) and modified maximum likelihood estimator (MMLE) using the Estimated Relative Error (ERE) as criterion of comparison. The result of a simulated study indicates that the EB estimator is more efficient than MME and MMLE. For application, the EB odds ratio estimation is implemented in AIDS-related data which the response was the self-reported number of times that respondents having a risky sexual partners, classified by gender. The estimation based EB also yields consistent result as those in simulation, resulting in smallest ERE when compared to MME and MMLE.INDEX TERMS Empirical Bayes, zero-inflated Poisson distribution, moments method estimation, modified maximum likelihood estimator.