  We study the problem of estimating the covariance parameters of a
one-dimensional Gaussian process with exponential covariance function under
fixed-domain asymptotics. We show that the weighted pairwise maximum likelihood
estimator of the microergodic parameter can be consistent or inconsistent. This
depends on the range of admissible parameter values in the likelihood
optimization. On the other hand, the weighted pairwise conditional maximum
likelihood estimator is always consistent. Both estimators are also
asymptotically Gaussian when they are consistent. Their asymptotic variances
are larger or strictly larger than that of the maximum likelihood estimator. A
simulation study is presented in order to compare the finite sample behavior of
the pairwise likelihood estimators with their asymptotic distributions. For
more general covariance functions, an additional inconsistency result is
provided, for the weighted pairwise maximum likelihood estimator of a variance
parameter.
