This paper provides a review of greedy pursuits for optimizing Volterra-based behavioral models structure and estimating its parameters. An experimental comparison of the digital predistortion (DPD) linearization performance achieved by these approaches for model-order reduction, such as compressive sampling matching pursuit (CoSaMP), subspace pursuit (SP), orthogonal matching pursuit (OMP), and the novel doubly OMP (DOMP), is presented. A benchmark of the techniques in the DPD of a commercial class AB power amplifier (PA) and a class J PA operating over a 15-MHz Long-Term Evolution (LTE) signal is presented, giving a clear overview of their pruning characteristics in terms of linearization indicators and regressor selection capabilities. In addition, the benchmark is run in a cross-validation scheme by identifying the DPD with a 30-MHz 5G-new radio (NR) signal and validating with the same signal and a 20-MHz multicarrier wideband code division multiple access (WCDMA) signal. The DOMP is shown to be a promising technique since it achieves an enhanced model-order reduction for a similar linearization performance and precision.