Application of maximum likelihood and bootstrap methods to nonlinear curve‐fit problems in geochemistry

Abstract

We apply maximum likelihood and bootstrap methods to the assess parameter estimates and confidence intervals in nonlinear geochemical data analysis. We focus primarily on the regression of “double error” data, or x−y data pairs, where both variables are measurements subject to error (e.g., isochron fitting), but the methods we discuss are general and can be applied to any nonlinear regression. We review the differences between least squares and maximum likelihood methods and describe how maximum likelihood can be naturally extended to permit robust regressions that are relatively insensitive to outliers. We describe the bootstrap technique for the estimation of the standard error and confidence intervals of a model and show how it can be used to sidestep the mathematical intractability of estimating uncertainty in nonlinear regressions. We illustrate the application of these methods through examples using synthetic and real data and demonstrate how they can influence statistical perception of nonlinear parameter estimates. In particular, we demonstrate how the linearized methods that are commonly used to make uncertainty estimates can substantially underestimate model error, and we show how this problem can be addressed with bootstrap confidence intervals that automatically adjust to account for intrinsic asymmetry and non‐Gaussian behavior in a regression. We include computer code to implement the calculations we describe as supplements to this article.