Performance Analysis of the Least-Squares Estimator in Astrometry

Abstract

We characterize the performance of the widely used least-squares estimator in astrometry in terms of a comparison with the Cramér–Rao lower variance bound. In this inference context the performance of the leastsquares estimator does not offer a closed-form expression, but a new result is presented (Theorem 1) where both the bias and the mean-square-error of the least-squares estimator are bounded and approximated analytically, in the latter case in terms of a nominal value and an interval around it. From the predicted nominal value, we analyze how efficient the least-squares estimator is in comparison with the minimum variance Cramér–Rao bound. Based on our results, we show that, for the high signal-to-noise ratio regime, the performance of the least-squares estimator is significantly poorer than the Cramér–Rao bound, and we characterize this gap analytically. On the positive side, we show that for the challenging low signal-to-noise regime (attributed to either a weak astronomical signal or a noisedominated condition) the least-squares estimator is near optimal, as its performance asymptotically approaches the Cramér–Rao bound. However, we also demonstrate that, in general, there is no unbiased estimator for the astrometric position that can precisely reach the Cramér–Rao bound. We validate our theoretical analysis through simulated digital-detector observations under typical observing conditions. We show that the nominal value for the mean-square-error of the least-squares estimator (obtained from our theorem) can be used as a benchmark indicator of the expected statistical performance of the least-squares method under a wide range of conditions. Our results are valid for an idealized linear (one-dimensional) array detector where intrapixel response changes are neglected, and where flat-fielding is achieved with very high accuracy.