This article proposes an optimal and robust methodology for model selection. The model of interest is a parsimonious alternative framework for modeling the stochastic dynamics of mortality improvement rates introduced by Doukhan et al. (2017). The approach models mortality improvements using a random field specification with a given causal structure instead of the commonly used factor-based decomposition framework. It captures some well documented stylized facts of mortality behavior: dependencies among adjacent cohorts, the cohort effects, cross generations correlations and the conditional heteroskedasticity of mortality. Such a class of models is a generalization of the now widely used AR-ARCH models for univariate processes. The framework being general, Doukhan et al. (2017) investigate and illustrate a simple variant, called the three-level memory model. However, it is not clear which is the best parametrization to use for specific mortality uses. In this paper, we investigate the optimal model choice and parameter selection among potential and candidate models. More formally, we propose a methodology well-suited to such a random field able to select the best model in the sense that the model is not only correct but also most economical among all the correct models. Formally, we show that a criterion based on a penalization of the log-likelihood, e.g. the using the Bayesian Information Criterion, is consistent. Finally, we investigate the methodology based on Monte-Carlo experiments as well as real-world datasets.