  The availability of large datasets requires an improved view on statistical
laws in complex systems, such as Zipf's law of word frequencies, the
Gutenberg-Richter law of earthquake magnitudes, or scale-free degree
distribution in networks. In this paper we discuss how the statistical analysis
of these laws are affected by correlations present in the observations, the
typical scenario for data from complex systems. We first show how standard
maximum-likelihood recipes lead to false rejections of statistical laws in the
presence of correlations. We then propose a conservative method (based on
shuffling and under-sampling the data) to test statistical laws and find that
accounting for correlations leads to smaller rejection rates and larger
confidence intervals on estimated parameters.
