  When the spatial sample size is extremely large, which occurs in many
environmental and ecological studies, operations on the large covariance matrix
are a numerical challenge. Covariance tapering is a technique to alleviate the
numerical challenges. Under the assumption that data are collected along a line
in a bounded region, we investigate how the tapering affects the asymptotic
efficiency of the maximum likelihood estimator (MLE) for the microergodic
parameter in the Mat\'ern covariance function by establishing the fixed-domain
asymptotic distribution of the exact MLE and that of the tapered MLE. Our
results imply that, under some conditions on the taper, the tapered MLE is
asymptotically as efficient as the true MLE for the microergodic parameter in
the Mat\'ern model.
