论文标题
标准最小二乘方法的轨道重建方法的不恰当性证明
Proofs of non-optimality of the standard least-squares method for track reconstructions
论文作者
论文摘要
它是统计的标准标准，是定义最佳估计值的标准标准。因此，最佳性证明了竞争估计差异方差之间的不平等。此处证明的不平等现象不利于标准最小二乘估计器。估计器之间的不平等与Cramer，Rao和Frechet的名称有关。这些不平等的标准演示需要概率函数的非常特殊的分析特性，该概率函数在全球范围内表示为常规模型。这些限制条件过于限制，无法处理轨道拟合中的现实问题。 Gaussian分布对Cramer-Rao-Frechet不平等的异质模型进行了以前的扩展。与标准最小二乘法相比，这些示威证明了异质模型的优越性。但是，高斯分布是所需的常规模型的典型成员。取而代之的是，在跟踪器检测器中遇到的现实概率分布与高斯分布非常不同。因此，要拥有一组良好的不平等，必须超越对常规模型的局限性。本文的目的是证明对概率不规则模型的最小二乘估计量的不平等，这明确排除了Cramer-rao-frechet示范。将考虑直线和抛物线轨道的估计器。最后一部分涉及使用最佳估计器和标准（非最佳）估计器重建的简化异质轨道模型的分布形式。这些不同估计器的分布之间的比较表明，标准最小二乘估计器的分辨率损失很大。
It is a standard criterium in statistics to define an optimal estimator the one with the minimum variance. Thus, the optimality is proved with inequality among variances of competing estimators. The inequalities, demonstrated here, disfavor the standard least squares estimators. Inequalities among estimators are connected to names of Cramer, Rao and Frechet. The standard demonstrations of these inequalities require very special analytical properties of the probability functions, globally indicated as regular models. These limiting conditions are too restrictive to handle realistic problems in track fitting. A previous extension to heteroscedastic models of the Cramer-Rao-Frechet inequalities was performed with Gaussian distributions. These demonstrations proved beyond any possible doubts the superiority of the heteroscedastic models compared to the standard least squares method. However, the Gaussian distributions are typical members of the required regular models. Instead, the realistic probability distributions, encountered in tracker detectors, are very different from Gaussian distributions. Therefore, to have well grounded set of inequalities, the limitations to regular models must be overtaken. The aim of this paper is to demonstrate the inequalities for least squares estimators for irregular models of probabilities, explicitly excluded by the Cramer-Rao-Frechet demonstrations. Estimators for straight and parabolic tracks will be considered. The final part deals with the form of the distributions of simplified heteroscedastic track models reconstructed with optimal estimators and the standard (non-optimal) estimators. A comparison among the distributions of these different estimators shows the large loss in resolution of the standard least-squares estimators.
