The effect of measurement errors on estimating power law distribution parameters and implications for measuring the IMF

Power law distributions are commonly found in astronomy. The masses of HI clouds, molecular clouds, and stars are frequently approximated as power law distributions. The Initial Mass Function (IMF) for stellar masses is normally approximated as a power law with slope or alpha of 2.35. The core mass function (CMF) is also of interest, as the discrepancy between the distribution of stars and cores could tell us how cores develop into stars. Important published results conclude that the estimated alpha of the CMF is different from the slope of the IMF. These estimates, however, neglect the inherent biases in the methods of fitting a power law distribution to data, and the effect of measurement errors on the estimate of alpha. In this work, we investigate the bias in three different methods: Maximum Likelihood Estimation of alpha as in Clauset 07, fit a regression line to cumulative distribution function of the data, and another Maximum Likelihood estimator for the convolved power law distribution found in Koen 09, which directly incorporates the measurement errors into the power law distribution. We considered two kinds of error, flat (changing the sigma of the error distribution by a constant for all masses in distribution) and proportional(changing sigma of the error distribution by by some fraction of the mass), and how incorporating these errors biased the estimated alpha for each of the methods. Fit a line to CDF method underestimates alpha, while the Clauset method overestimates alpha. The Clauset method has less bias in the proportional case, but in performs worse than the fit a line to CDF method in the flat error case. This effect is because the Clauset method estimated alpha and the power law lower bound on the power law distribution. For increasing amounts of flat errors, the distribution looks less and less like a power law distribution, so the estimated alpha worsens because the data is no longer well approximated by a power law. We suggest a method on finding bias adjustments given an estimated alpha and method of fitting.