Optimal guaranteed estimation methods for the Cox–Ingersoll–Ross models

In this paper, we study parameter estimation problems for the Cox–Ingersoll–Ross (CIR) processes. For the first time for such models, sequential estimation procedures are proposed. In the non-asymptotic setting, the proposed sequential procedures provide the estimation with non-asymptotic fixed mean square accuracy. For the scalar parameter estimation problems non-asymptotic normality properties for the proposed estimators are established even in the cases when the classical non-sequential maximum likelihood estimators cannot be calculated. Moreover, the Laplace transformations for the mean observation durations are obtained. In the asymptotic setting, the limit forms for the mean observation durations are found and it is shown that the constructed sequential estimators uniformly converge in distribution to normal random variables. Then using the Local Asymptotic Normality (LAN) property, it is obtained asymptotic sharp lower bound for the minimax risks in the class of all sequential procedures with the same mean observation duration and as a consequence, it is established that the proposed sequential procedures are optimal in the minimax sense in this class.