Influence of design on the rate of convergence to normality in L₁ regression

We provide a concise account of the influence of design variables on the convergence rate in an L₁ regression problem. In particular, we show that the convergence rate may be characterized precisely in terms of third and fourth moments of the design variables. This result leads to necessary and sufficient conditions on the design for the Berry-Esseen rate to be achieved. We also show that a moment condition on the error distribution is necessary and sufficient for a nonuniform Berry-Esseen theorem, and that an Edgeworth expansion is possible if the design points are not too clumped.