标题： 天文测量中最小二乘估计器的性能分析 显示英文标题

标题： Performance analysis of the Least-Squares estimator in Astrometry

摘要： 我们通过与Cramér-Rao下限方差界的比较来表征天文学中最常用的最小二乘估计器的性能。 在此推断背景下，最小二乘估计器的性能没有封闭形式的表达式，但提出了一项新结果（定理1），其中最小二乘估计器的偏倚和均方误差都被解析地限制和近似，在后一种情况下以一个标称值及其周围的区间表示。 从预测的标称值出发，我们分析了最小二乘估计器相对于最小方差Cramér-Rao界的有效性。 基于我们的结果，我们表明，在高信噪比的情况下，最小二乘估计器的性能明显不如Cramér-Rao界，并且我们对该差距进行了分析。 从积极的一面来看，我们证明，在具有挑战性的低信噪比情况（归因于较弱的天文信号或噪声占主导的情况）下，最小二乘估计器接近最优，因为其性能渐近达到Cramér-Rao界。 然而，我们也证明，一般来说，不存在能够精确达到Cramér-Rao界的无偏天文学位置估计器。 我们通过在典型观测条件下模拟数字探测器观测来验证我们的理论分析。 我们展示了最小二乘估计器的均方误差的标称值（由我们的定理得出）可以作为最小二乘法在广泛条件下的预期统计性能的基准指标。 我们的结果适用于理想化的线性（一维）阵列探测器，其中忽略了像素内响应的变化，并且通过非常高的精度实现了平场校正。 显示英文摘要

摘要： We characterize the performance of the widely-used least-squares estimator in astrometry in terms of a comparison with the Cramer-Rao lower variance bound. In this inference context the performance of the least-squares 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 is the least-squares estimator in comparison with the minimum variance Cramer-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 Cramer-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 noise-dominated condition) the least-squares estimator is near optimal, as its performance asymptotically approaches the Cramer-Rao bound. However, we also demonstrate that, in general, there is no unbiased estimator for the astrometric position that can precisely reach the Cramer-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 intra-pixel response changes are neglected, and where flat-fielding is achieved with very high accuracy.