  We consider covariance parameter estimation for a Gaussian process under
inequality constraints (boundedness, monotonicity or convexity) in fixed-domain
asymptotics. We address the estimation of the variance parameter and the
estimation of the microergodic parameter of the Mat\'ern and Wendland
covariance functions. First, we show that the (unconstrained) maximum
likelihood estimator has the same asymptotic distribution, unconditionally and
conditionally to the fact that the Gaussian process satisfies the inequality
constraints. Then, we study the recently suggested constrained maximum
likelihood estimator. We show that it has the same asymptotic distribution as
the (unconstrained) maximum likelihood estimator. In addition, we show in
simulations that the constrained maximum likelihood estimator is generally more
accurate on finite samples. Finally, we provide extensions to prediction and to
noisy observations.
