Parameter estimation in complex models arising in real data applications is a topic which still attracts a lot of interest. In this article, we study a specific data and parameter augmentation method which gives us the opportunity to estimate more easily the parameters of the initial model. For this reason, the notion of Gaussian randomization of a model with respect to some of its parameters is introduced. The initial model can be regarded as a submodel of the resulting extended incomplete data model. Under the assumption that the initial model has a unique maximum likelihood estimator (MLE) and that the likelihood function is continuous we prove that the extended model has a unique MLE with common values for the parameters of the MLE which correspond to the initial model. We also prove the reverse direction. Moreover, an appropriate stochastic version of an EM (Expectation-Maximization) algorithm is suggested to make parameter estimation feasible. In particular, we describe how the regularized particle filter of can be used in this frequentist-based approach to perform the Monte Carlo E-step at each iteration of the stochastic EM algorithm. This regularized version is particularly adapted to the framework of Gaussian randomization since the last iterations of the EM algorithm are characterized by low variance in the parameter distributions. A toy example with available analytic solutions, a synthetic example and a real data application with scarce observations to the LNAS (Log-Normal Allocation and Senescence) model of sugar beet growth are presented to highlight some theoretical and practical aspects of the proposed methodology.