The Euler–Riemann ζ function and the estimation of the power-law exponent

Power-law distributions are widely used in statistical analysis of extreme events and complex systems. For the commonly used power-law distributions, the variances do not exist. This makes the error analysis of point estimation and the construction of interval estimation of exponent parameter more difficult. Based on Euler product formula, this paper gives a summation formula of derivative of logarithmic Riemann ζ function and provides a more convenient method for calculating maximum likelihood estimation. By compressed transformation, the variance is ensured to be finite and the maximum likelihood estimation of the exponent is equivalent to its moment estimation. So a method of interval estimation based on the central limit theorem is proposed. In addition, this paper also gives an interval estimation method based on the acceptance domain of likelihood ratio test. The advantages of the proposed methods are showed in synthetic power-law data and illustrated with applications in a set of empirical data.