Statistical Methods for Jointly Estimating the Decay Constant of 40K and the Age of a Dating Standard

To simultaneously evaluate the decay constant of 40K (λ) and the age of a standard (tstd) using isotopic data from geologic materials, we applied a series of statistical methods. The problem of estimating the most probable intercept of many nonlinear curves in λ and tstd space is formulated by an errors-in-variables nonlinear regression model. Then a maximum likelihood method is applied to the model for a point estimate, which is equivalent to the nonlinear least square method when measurement error distributions are Gaussian. Uncertainties and confidence regions of the estimates can be approximated using three methods: the asymptotic normal approximation, the parametric bootstrap method and Bonferroni confidence regions. Five pairs of published data for samples with ages from 2 ka to 4.5 Ga were used to estimate λ and the age of Fish Canyon sanidine (tFCs). The statistical procedure yields most probable estimates of λ (5.4755 ± 0.0170 × 10−10 (1σ)/year) and tFCs (28.269 ± 0.0661 (1σ) Ma) which are in between previously published values. These results indicate the power of our approach to provide improved constraints on these parameters, although the preliminary nature of some of the input data require further review before the values can be adopted.