  Suppose that $k$ series, all having the same autocorrelation function, are
observed in parallel at $n$ points in time or space. From a single series of
moderate length, the autocorrelation parameter $\beta$ can be estimated with
limited accuracy, so we aim to increase the information by formulating a
suitable model for the joint distribution of all series. Three Gaussian models
of increasing complexity are considered, two of which assume that the series
are independent. This paper studies the rate at which the information for
$\beta$ accumulates as $k$ increases, possibly even beyond $n$. The profile log
likelihood for the model with $k(k+1)/2$ covariance parameters behaves
anomalously in two respects. On the one hand, it is a log likelihood, so the
derivatives satisfy the Bartlett identities. On the other hand, the Fisher
information for $\beta$ increases to a maximum at $k=n/2$, decreasing to zero
for $k\ge n$. In any parametric statistical model, one expects the Fisher
information to increase with additional data; decreasing Fisher information is
an anomaly demanding an explanation.
