An observer performing a detection task analyzes an image and produces a single number, a test statistic, for that image. This test statistic represents the observers "confidence" that a signal (e.g., a tumor) is present. The linear observer that maximizes the test-statistic SNR is known as the Hotelling observer. Generally, computation of the Hotelling SNR, or Hotelling trace, requires the inverse of a large covariance matrix. Recent developments have resulted in methods for the estimation and inversion of these large covariance matrices with relatively small numbers of images. The estimation and inversion of these matrices is made possible by a covariance-matrix decomposition that splits the full covariance matrix into an average detector-noise component and a background-variability component. Because the average detector-noise component is often diagonal and/or easily estimated, a full-rank, invertible covariance matrix can be produced with few images. We have studied the bias of estimates of the Hotelling trace using this decomposition for high-detector-noise and low-detector-noise situations. In extremely low-noise situations, this covariance decomposition may result in a significant bias. We will present a theoretical evaluation of the Hotelling-trace bias, as well as extensive simulation studies.