Bias reduction and goodness-of-fit in discrete power-law models

{Likelihood-based modelling and Monte Carlo simulation are used to develop and assess Fisher-penalized inference, goodness-of-fit procedures, and piecewise modelling for discrete power-law distribution.} Statistical inference for discrete power-law models remains challenging due to finite-sample bias in the scaling exponent, limitations in goodness-of-fit assessment, and the inability of the standard model to accommodate more than one scaling regime. In this paper, we develop a unified framework addressing bias reduction and goodness-of-fit testing for both the tail-based power-law model and a piecewise extension that describes the entire distribution. For the tail-based model, we introduce Fisher-penalized maximum likelihood estimators of the scaling exponent, adapting Firth-type bias reduction to the discrete power-law likelihood through a penalty derived from the Fisher information. For the piecewise extension, which incorporates change points, bias in the segment-specific exponents is reduced via bootstrap correction. In both settings, we investigate goodness-of-fit procedures for testing power-law behaviour, comparing the Kolmogorov–Smirnov, Cramér–von Mises, Watson, and Anderson–Darling statistics through Monte Carlo simulations. The results reveal that the Fisher-penalized and bootstrap-corrected estimators outperform their classical maximum likelihood counterparts, and that the Watson statistic achieves the most favourable balance between type I error control and power for both models, whereas the Kolmogorov–Smirnov statistic, commonly employed for this purpose, frequently fails to preserve the nominal significance level. Applications to network degree sequences and word-frequency data illustrate the practical advantages of the proposed methodology. The Watson test recovers several hundred power-law-compatible degree sequences rejected by the Kolmogorov-Smirnov test, and the piecewise model provides an adequate whole-distribution fit that outperforms Poisson, discrete log-normal, and discrete exponential competitors in most cases considered, although a whole-distribution alternative achieves a marginally better fit in one application.