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

TL;DR

This paper reviews and introduces bias-free estimators for power-law parameters, discusses graphical and statistical goodness-of-fit tests, and applies these methods to astronomical star data.

Contribution

It provides new bias-free estimators for the exponent and upper limit of power-law distributions, along with software tools and goodness-of-fit tests.

Findings

Bias-free estimators improve parameter accuracy

Graphical methods aid in data interpretation

Goodness-of-fit tests validate power-law models

Abstract

Many objects studied in astronomy follow a power law distribution function, 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 distribution function, 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. The software packages are made available as downloads. 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. (abridged)