function representation is favored over that of the pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended. Estimating the exponent from empirical data There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers. Some of the most reliable techniques are often based on the method of maximum likelihood. Alternative methods are often based on making a linear regression on either the log–log probability, the log–log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent. Maximum likelihood For real-valued, independent and identically distributed data, we fit a power-law distribution of the form to the data , where the coefficient is included to ensure that the distribution is normalized. Given a choice for , the log likelihood function becomes: The maximum of this likelihood is found by differentiating with respect to parameter , setting the result equal to zero. Upon rearrangement, this yields the estimator equation: where are the data points . This estimator exhibits a small finite sample-size bias of order , which is small when n > 100. Further, the standard error of the estimate is . This estimator is equivalent to the popular Hill estimator from quantitative finance and extreme value theory. For a set of n integer-valued data points , again where each , the maximum likelihood exponent is the solution to the transcendental equation where is the incomplete zeta function. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa. Further, both of these estimators require the choice of . For functions with a non-trivial function, choosing too small produces a significant bias in , while choosing it too large increases the uncertainty in , and reduces the statistical power of our model. In general, the best choice of depends strongly on the particular form of the lower tail, represented by above. More about these methods, and the conditions under which they can be used, can be found in . Further,