  In this work the primary objective is to maximize the precision of the
maximum likelihood estimate in a linear regression model through the efficient
design of the experiment. One common measure of precision is the unconditional
mean square error. Unconditional mean square error has been a primary motivator
for optimal designs; commonly, defined as the design that maximizes a concave
function of the expected Fisher information. The inverse of expected Fisher
information is asymptotically equal to the mean square error of the maximum
likelihood estimate. There is a substantial amount of existing literature that
argues the mean square error conditioned on an appropriate ancillary statistic
better represents the precision of the maximum likelihood estimate. Despite
evidence in favor of conditioning, limited effort has been made to find designs
that are optimal with respect to conditional mean square error. The inverse of
observed Fisher information is a higher order approximation of the conditional
mean square error than the inverse of expected Fisher information [Efron and
Hinkley (1978)]. In light of this, a more relevant objective is to find designs
that optimize observed Fisher information. Unlike expected Fisher information,
observed Fisher information depends on the observed data and cannot be used to
design an experiment completely in advance of data collection. In a sequential
experiment the observed Fisher information from past observations is available
to inform the design of the next observation. In this work an adaptive design
that incorporates observed Fisher information is proposed for linear regression
models. It is shown that the proposed design is more efficient, at the limit,
than any fixed design, including the optimal design, with respect to
conditional mean square error.
