Parametric estimation for the mean of a Gaussian process by the method of sieves

In this paper, we apply Grenader's method of sieves to the problem of estimation of the infinite dimensional mean parameter of a Gaussian random vector with values in a real separable Banach space. We use an increasing sequence of natural sieves in terms of geometric properties of the parameter space, on which we maximize the likelihood function. Sufficient conditions on the growth of the sieves are given in order that the sequence of restricted maximum likelihood estimators of the unknown mean is consistent when the sample size tends to infinity. Exponential rates of convergence in the topology of the parameter space are obtained. Applications to some examples are also discussed.