2022 Volume 7 Issue 3 Pages 137-148
Conway and Maxwell derived the Conway-Maxwell (COM)-Poisson distribution as generalization of the Poisson distribution. This distribution has been used in survival analysis. The probability mass function (pmf) of this distribution contains a normalizing constant expressed as sum of infinite series and therefore, not only the computation of the distribution but also the parameter estimation for the COM-Poisson is difficult. To remedy this problem, several methods have been appeared in the literature such as the methods based on Laplace approximation and linear regression. However, it is pointed out that the approximation accuracy of the Laplace approximation is poor, and the regression method cannot be applied if there are no covariates.
In this paper, we propose a new method of parameter estimation for the COM-Poisson using the conditional likelihood functions in the COM-Poisson distribution. The key idea of the proposed method is to use the conditional likelihood functions, which does not have the complicated normalizing constant. We further prove that the estimates of all two parameters always exist uniquely and a conditional likelihood function of the shape parameter is a log-concave function. Through Monte Carlo simulations, we further show that the proposed method performs better than the existing method in terms of bias and root mean squared error (RMSE). In an illustrative example, we fit the COM-Poisson model to the real data set of carton by our proposed method.
