  Two canonical problems in geostatistics are estimating the parameters in a
specified family of stochastic process models and predicting the process at new
locations. A number of asymptotic results addressing these problems over a
fixed spatial domain indicate that, for a Gaussian process with Mat\'ern
covariance function, one can fix the range parameter controlling the rate of
decay of the process and obtain results that are asymptotically equivalent to
the case that the range parameter is known. In this paper we show that the same
asymptotic results can be obtained by jointly estimating both the range and the
variance of the process using maximum likelihood or maximum tapered likelihood.
Moreover, we show that intuition and approximations derived from asymptotic
arguments using a fixed range parameter can be problematic when applied to
finite samples, even for moderate to large sample sizes. In contrast, we show
via simulation that performance on a variety of metrics is improved and
asymptotic approximations are applicable for smaller sample sizes when the
range and variance parameters are jointly estimated. These effects are
particularly apparent when the process is mean square differentiable or the
effective range of spatial correlation is small.
