Two step estimations via the Dantzig selector for models of stochastic processes with high-dimensional parameters

We consider the sparse estimation for stochastic processes with possibly infinite-dimensional nuisance parameters, by using the Dantzig selector which is a sparse estimation method similar to $Z$-estimation. When a consistent estimator for a nuisance parameter is obtained, it is possible to construct an asymptotically normal estimator for the parameter of interest under appropriate conditions. Motivated by this fact, we establish the asymptotic behavior of the Dantzig selector for models of ergodic stochastic processes with high-dimensional parameters of interest and possibly infinite-dimensional nuisance parameters. Moreover, we construct an asymptotically normal estimator by the two step estimation with help of the variable selection through the Dantzig selector and a consistent estimator of the nuisance parameter. Applications to ergodic time series models including integer-valued autoregressive models and ergodic diffusion processes are presented.