Lasso Maximum Likelihood Estimation of Parametric Models with Singular Information Matrices

An information matrix of a parametric model being singular at a certain true value of a parameter vector is irregular. The maximum likelihood estimator in the irregular case usually has a rate of convergence slower than the root n-rate in a regular case. We propose to estimate such models by the adaptive lasso maximum likelihood and propose an information criterion to select the involved tuning parameter. We show that the penalized maximum likelihood estimator has the oracle properties. The method can implement model selection and estimation simultaneously and the estimator always has the usual root n-rate of convergence.