System identification (SI), especially from small samples, is a challenging problem and of interest in several applications. Standard prediction-error minimization methods (PEM), under these conditions, generally result in estimates with higher variance. Moreover, in the identification of parametric models, one often needs prior knowledge of the input-output delay, obtaining estimates of which, is not possible using classical methods when the delay is either comparable or greater than the sample size. In this work, we develop a compressed sensing (CS)-based method for identifying sparse equation-error models that includes both auto-regressive eXogenous (ARX) and AR moving average eXogenous (ARMAX) structures with large delays, small orders and small delays with large orders, but with missing coefficients. The outcome is an iterative basis pursuit de noising (IBPDN) algorithm for solving non-linear CS problems. In addition, we propose a semi-rigorous method to lower the mutual coherence of the regressor matrix so as to obtain lower variance parameter estimates with the CS techniques. Errors in parameter estimates are computed using the bootstrapping method. Simulation studies on three diverse examples are presented to demonstrate the efficacy of the proposed methodology. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.