Fitting power-law distributions to data with measurement errors
Abstract
If X, which follows a power-law distribution, is observed subject to Gaussian measurement
error e, thenX+e is distributed as the convolution of the power-lawand Gaussian distributions.
Maximum-likelihood estimation of the parameters of the two distributions is considered.
Large-sample formulae are given for the covariance matrix of the estimated parameters, and
implementation of a small-sample method (the jackknife) is also described. Other topics dealt
with are tests for goodness of fit of the posited distribution, and tests whether special cases (no
measurement errors or an infinite upper limit to the power-law distribution) may be preferred.
The application of the methodology is illustrated by fitting convolved distributions to masses
of giant molecular clouds in M33 and the Large Magellanic Cloud (LMC), and to HI cloud
masses in the LMC.