The stochastic EM algorithm replaces the E-step with a Monte Carlo approximation, trading monotonicity for the potential to escape local maxima. Estimation techniques include averaging the tail of the chain, and choosing the value in the chain associated with the largest likelihood value. We demonstrate that the latter estimator diverges from the maximum likelihood estimate with high probability as the dimensionality of the parameter increases, but that it is also more precise in terms of chain length when the parameter is a scalar. Based on these findings, we propose a new estimator which we prove achieves this same level of precision for inference of multidimensional parameters. Simulation studies demonstrate the benefits of the proposed estimator when compared to topical approaches. Some discussion is given on limitations of the proposed estimator and potential avenues for increasing applicability are considered.