In this paper we consider maximum likelihood estimation of the rate constant for stochastic rth-order reactions based on observation of the level of the system at time t > 0. Conditions are found for which the waiting time until the nth. reaction event is asymptotically normal, as the initial number of molecules and the number of reaction events become large. This distributional result is used to derive an approximate estimator which is shown for the second-order case to be close to the exact maximum likelihood estimate over a wide range of percentage completion.

In this paper we derive the probability distributions of the number of molecules and of the lifetime of a molecule in a stochastic rth-order system by direct evaluation of probabilities, avoiding the use of differential-difference equations. Maximum likelihood estimation of the rate constant is based on an observation of the level of the system at time t > 0. We find the asymptotic solution of the likelihood equation for a large initial number of molecules. By comparison with the numerical solution of the likelihood equation, the asymptotic estimator is shown to be a satisfactory approximation for second order reactions which are far from completion.