New and fast closed-form efficient estimators for the negative multinomial distribution

The negative multinomial (NM) distribution is of interest in various application studies. Based on closed-form n-consistent estimators, new and fast closed-form efficient estimators are proposed for the NM distribution. The theorem applied to derive the new estimators guarantees two important properties of the new closed-form efficient estimators: asymptotic efficiency and normality. The new closed-form efficient estimators are denoted as MLE-CEs, because the asymptotic distribution is the same as that of the maximum likelihood estimators (MLEs). Simulation studies suggest that the MLE-CE performs similarly to its MLE. The estimated accuracies of the MLE and MLE-CE are generally better than the method of moments estimator (MME) for relatively large 𝒑 values. The MLE-CE is 10–30 times faster than the MLE, especially for large sample sizes, which is good for the big data era. Considering the estimated accuracy and computing time, the MLE-CE is recommended for large 𝒑 values, whereas the MME is recommended for other conditions.