Summary: In this paper, we focus on applying restricted maximum likelihood to the linear mixed model with auto-correlated error structure under the elliptical distribution class. Maximum likelihood estimation under both linear equality and inequality constraints on location parameters are considered. Three asymptotically equivalent tests are also constructed for the interest parameters under restrictions. Monte Carlo simulations are conducted to evaluate the effectiveness and robustness of the new methodology. An illustrative example for the cholesterol levels from the Framingham heart study is reported.