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.
A detailed discussion of the calculation of the "best straight line" by the method of least squares is given. The most general solution is found and the conditions under which certain previously derived special solutions are valid are clearly stated. The "best" slope is shown to be given by the solution of the "Least-Squares Cubic". An example is given to illustrate the method. It is shown that the best slope is not necessarily bounded by values found from the regressions of x on y and y on x.
Argon-40-argon-39 single-crystal dating of young (5000 to 30,000 years ago) volcanic ash layers erupted from the Mono Craters, California, shows that the method can yield meaningful ages in Holocene tephra. Because of ubiquitous xenocrystic contamination, the data do not form isochrons but plot in wedge-shaped regions on an argon isotopic diagram. The upper boundary of the region is an isochron matching the 14C-derived age of the eruption. Such contamination-related patterns may be common in dating young materials by the single-crystal method. Argon dating by this method can help refine the time scale of physical and biological evolution over the past 100,000 years.
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