Unified equations for the slope, intercept, and standard errors of the best straight line
Unified equations for the slope, intercept, and standard errors of the best straight line
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DOI：
10.1119/1.1632486
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发表时间：
2004-03-01
影响因子：
0.9
通讯作者：
Delgado, JD
中科院分区：
文献类型：
--
作者：
York, D;Evensen, NM;Delgado, JD
人们早就认识到，将最佳直线拟合到具有正态分布误差的数据点的最小二乘估计方法与最大似然估计方法一样，对直线的斜率和截距产生相同的结果。我们表明，与以前的理解相反，这两种方法也给出了相同的结果的标准误差的斜率和截距，提供的最小二乘估计表达式进行评估，在最小二乘调整点，而不是在观察点，因为传统上已经完成。当x和y观测值都受到随点变化的相关误差的影响时，标准误差的这种统一成立。文献中所有已知的正确的回归解，包括各种特殊情况，都可以从原始的约克方程推导出来。我们提出了一组简洁的斜率、截距和新统一的标准误差方程。(C)2004年美国物理教师协会。
It has long been recognized that the least-squares estimation method of fitting the best straight line to data points having normally distributed errors yields identical results for the slope and intercept of the line as does the method of maximum likelihood estimation. We show that, contrary to previous understanding, these two methods also give identical results for the standard errors in slope and intercept, provided that the least-squares estimation expressions are evaluated at the least-squares-adjusted points rather than at the observed points as has been done traditionally. This unification of standard errors holds when both x and y observations are subject to correlated errors that vary from point to point. All known correct regression solutions in the literature, including various special cases, can be derived from the original York equations. We present a compact set of equations for the slope, intercept, and newly unified standard errors. (C) 2004 American Association of Physics Teachers.