Estimators for the exponent and upper limit, and goodness-of-fit tests for (truncated) power-law distributions

Abstract

Many objects studied in astronomy follow a power-law distribution function (DF), for example the masses of stars or star clusters. A still used method by which such data is analysed is to generate a histogram and fit a straight line to it. The parameters obtained in this way can be severely biased, and the properties of the underlying DF, such as its shape or a possible upper limit, are difficult to extract. In this work, we review techniques available in the literature and present newly developed (effectively) bias-free estimators for the exponent and the upper limit. Furthermore, we discuss various graphical representations of the data and powerful goodness-of-fit tests to assess the validity of a power law for describing the distribution of data. As an example, we apply the presented methods to the data set of massive stars in R136 and the young star clusters in the Large Magellanic Cloud. For R136 we confirm the result of Koen of a truncated power law with a bias-free estimate for the exponent of 2.20 ± 0.78/2.87 ± 0.98 (where the Salpeter-Massey value is 2.35) and for the upper limit of 143 ± 9/163 ± 9 M ⊙, depending on the stellar models used. The star clusters in the Large Magellanic Cloud (with ages up to 107.5 yr) follow a truncated power-law distribution with exponent 1.62 ± 0.06 and upper limit 68 ± 12 × 103 M⊙. Using the graphical data representation, a significant change in the form of the mass function below 102.5 M⊙ can be detected, which is likely caused by incompleteness in the data. © 2009 RAS.