Urban scaling with censored data

In the realm of urban science, scaling laws are essential for understanding the relationship between city population and urban features, such as socioeconomic outputs. Ideally, these laws would be based on complete datasets; however, researchers often face challenges related to data availability and reporting practices, resulting in datasets that include only the highest observations of the urban features (top-k). A key question that emerges is: Under what conditions can an analysis based solely on top-k observations accurately determine whether a scaling relationship is truly superlinear or sublinear? To address this question, we conduct a numerical study that explores how relying exclusively on reported values can lead to erroneous conclusions, revealing a selection bias that favors sublinear over superlinear scaling. In response, we develop a method that provides robust estimates of the minimum and maximum potential scaling exponents when only top-k observations are available. We apply this method to two case studies involving firearm violence, a domain notorious for its suppressed datasets, and we demonstrate how this approach offers a reliable framework for analyzing scaling relationships with censored data.