This paper describes a novel speech coding concept created by in-troducing sparsity constraints in a linear prediction scheme both on the residual and on the prediction vector. The residual is efficiently encoded using well known multi-pulse excitation procedures due to its sparsity. A robust statistical method for the joint estimation of the short-term and long-term predictors is also provided by exploiting the sparse characteristics of the predictor. Thus, the main purpose of this work is showing that better statistical modeling in the context of speech analysis creates an output that offers better coding prop-erties. The proposed estimation method leads to a convex optimiza-tion problem, which can be solved efficiently using interior-point methods. Its simplicity makes it an attractive alternative to com-mon speech coders based on minimum variance linear prediction.