A Maximum Likelihood Method for Power Law Distributions That Does Not Break Down When the Slope Is Close to Unity

Abstract

A general maximum likelihood estimation (MLE) method is given to analyze experimental data with a power law form with any power exponent which does not break down for a power close to −1. It contrasts thereby with a standard procedure that does. It can be extended to a power law with an exponential tail and more generally to other distribution forms. Inasmuch as the theoretical value of the power for dye-sensitized charge recombination in semiconductors systems, and for certain charge injection, is −1 (Chen, W.; Marcus, R. A., J. Phys. Chem. C, accepted), the present correction to the current MLE method has immediate application to the data in these systems, but it is equally applicable to other systems, regardless of whether the power is −1.