Multiport reflectometers permit the determination of both the amplitude and phase of coherent microwave signals from the measurement of power alone. A feature of the multiport technique is that it creates a set of mutually dependent estimates for the value of the unknown reflection coefficient Γ. The paper concerns a statistical method which generates a best linear unbiased estimator for reflection coefficient Γ, based on the criterion of minimum variance; such an estimator has many useful general properties. In addition to providing a single best estimator for Γ, the method also gives its standard deviation, a direct measure of its probable error, and such a measure of system performance can constitute a valuable diagnostic in multiport-reflectometer networks.