Comparing Distributions is a fundamental aspect of statistical data analysis, extending beyond traditional goodness-of-fit testing. While goodness-of-fit tests formally assess hypotheses in one-sample and K-sample scenarios, a broader approach to comparing distributions provides a more insightful and versatile toolkit for researchers and practitioners. This comprehensive methodology incorporates graphical techniques and estimation methods, offering a deeper understanding of data and enhancing the interpretability of results.
This approach emphasizes the “informative” nature of statistical procedures. An informative procedure goes beyond simply rejecting a null hypothesis; it elucidates why the hypothesis is rejected. By integrating diverse methods, we reveal the underlying reasons for differences or similarities between datasets, moving beyond a simple binary outcome of acceptance or rejection. Despite the varied historical development of these statistical tools, a unified theoretical framework highlights the inherent connections and shared principles among them.
The study of comparing distributions can be broadly divided into two key areas. First, methods designed for the one-sample problem, where a single dataset is analyzed against a theoretical distribution or known standard. Second, the K-sample problem, which involves comparing multiple datasets to identify differences or similarities in their underlying distributions. The techniques applicable to K-sample problems are particularly relevant for statisticians engaged in comparative studies across diverse fields.
A robust analysis of distribution comparisons encompasses a range of techniques, including graphical explorations, hypothesis testing, model selection, and density estimation. These methods draw upon parametric, semiparametric, and nonparametric statistical theory, presented with a balanced approach that emphasizes both theoretical foundations and practical intuition. Heuristics and intuitive explanations complement the theoretical rigor, making these powerful tools accessible to a wider audience. Practical application is further emphasized through numerous data examples, all analyzed using the author-developed cd R-package. Each example includes readily accessible R-code, enabling readers to replicate analyses and apply these techniques to their own data.
Given the broad applicability of these methods, “comparing distributions” is an essential component of any statistician’s toolkit. This approach is invaluable for researchers, graduate students, and PhD candidates seeking a solid foundation in goodness-of-fit testing and beyond. Practitioners and applied statisticians will also find significant value in the practical examples, R-code, and the emphasis on extracting meaningful insights from data comparisons. By moving beyond simple hypothesis rejection, comparing distributions empowers analysts to gain a richer, more informative understanding of their data.
