Statistical methods with empirical likelihood (EL) are appealing and ef- fective especially in conjunction with estimating equations for flexibly and adaptively incorporating data information. It is known that EL approaches en- counter difficulties when dealing with high-dimensional problems. To over- come the challenges, we begin our study with investigating high-dimensional EL from a new scope targeting at high-dimensional sparse model parame- ters. We show that the new scope provides an opportunity for relaxing the stringent requirement on the dimensionality of the model parameters. Moti- vated by the new scope, we then propose a new penalized EL by applying two penalty functions respectively regularizing the model parameters and the associated Lagrange multiplier in the optimizations of EL. By penalizing the Lagrange multiplier to encourage its sparsity, a drastic dimension reduction in the number of estimating equations can be achieved. Most attractively, such a reduction in dimensionality of estimating equations can be viewed as a selection among those high-dimensional estimating equations, resulting in a highly parsimonious and effective device for estimating high-dimensional sparse model parameters. Allowing both the dimensionalities of model pa- rameters and estimating equations growing exponentially with the sample size, our theory demonstrates that our new penalized EL estimator is sparse and consistent with asymptotically normally distributed nonzero components. Numerical simulations and a real data analysis show that the proposed penal- ized EL works promisingly.
