The use of X-ray pulsar signals appears to be a potential solution for autonomous deep space navigation. The main challenge in this kind of navigation is to estimate very precisely the initial phase of the pulse arriving at the detector. Consequently, a study of statistical performance is of the utmost interest. Previous studies indicate that in the performance of pulse phase estimators, the so-called threshold phenomenon arises when the observation time is below a critical limit. In this correspondence, to provide a prediction of the threshold position, the closed-form expressions of the lower bounds on the mean square error are derived and analyzed in both deterministic and Bayesian contexts. These bounds demand less computational cost than the classical Monte Carlo simulations. Numerical results show that the proposed bounds are able to predict the threshold location in both contexts.