New closed-form efficient estimators for the negative binomial distribution

The negative binomial (NB) distribution is of interest in various application studies. New closed-form efficient estimators are proposed for the two NB parameters, based on closed-form $$\sqrt{n}$$ n -consistent estimators. The asymptotic efficiency and normality of the new closed-form efficient estimators are guaranteed by the theorem applied to derive the new estimators. Since the new closed-form efficient estimators have the same asymptotic distribution as the maximum likelihood estimators (MLEs), these are denoted as MLE-CEs. Simulation studies suggest that the MLE-CE of dispersion parameter r performs better than its MLE and the method of moments estimator (MME) for some parameter ranges. The MLE-CE of the probability parameter p exhibits the best performance for relatively large p values, where the positive-definite expected Fisher information matrix exists. MLE performs better than MME in this parameter space. The MLE-CE is over 200 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, MLE-CE is recommended for small r values and large p values, whereas MME is recommended for other conditions.