Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this paper, a statistical regression method based on the maximum likelihood estimation considering the lower bound constraints and the heterogeneous variance of error structure, termed as CHVR, is proposed for urban scaling estimation. In the CHVR method, the heterogeneous properties of variance are explored and modeled in the form of a power-of-the-mean variance model. The maximum likelihood fitting method is supplemented to satisfy the lower bound constraints in empirical data. The CHVR method has been applied to estimating the scaling exponents of six urban attributes covering three scaling regimes in China and compared with two traditional methods. Method evaluations based on three different criteria validate that compared with both classical methods, the CHVR method is more effective and robust. Moreover, a statistical test and long-term variations of the parameter in the variance function demonstrate that the proposed heterogeneous variance function can not only describe the heterogeneity in empirical data adequately but also provide more meaningful urban information. Therefore, the CHVR method shows great potential to provide a valuable tool for effective urban scaling studies across the world and be applied to scaling law estimation in other complex systems in the future.
Keywords: China; complex systems; heterogeneity; scaling laws; urban systems.