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# Modern and classical k ‐sample omnibus tests

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The ‘k‐sample problem’ aims to detect statistical differences among multiple populations. A statistical test, capable of detecting any departure from the null hypothesis of ‘statistical equality’ such as the equality in distribution, is typically referred to as an ‘omnibus test.’ A short overview of historic developments and a detailed discussion of the more prominent state‐of‐the‐art techniques are presented with references to numerous sources and studies. Both classical and modern omnibus tests are systematically categorized in terms of seminal probabilistic and statistical concepts into tests that are based upon the empirical distribution, characteristic or kernel density function, etc. To demonstrate the strengths and weaknesses of each particular approach with regard to its statistical performance, applicability, computational complexity, and parameter tuning, eight representatively selected omnibus tests (along with the Kruskal–Wallis test) are numerically implemented and compared under various ‘archetypal’ scenarios. Recommendations are made accordingly along with a discussion of challenges and potential future research directions for this problem. WIREs Comput Stat 2018, 10:e1418. doi: 10.1002/wics.1418

• Statistical Models > Linear Models
• Statistical and Graphical Methods of Data Analysis > Nonparametric Methods
Probability density functions (PDF) and cumulative distribution functions (CDF) for two probability distributions.
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Probability density functions of $X3j3$, j3 = 1, …, n3, for various models.
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Statistical and Graphical Methods of Data Analysis > Nonparametric Methods
Statistical Models > Linear Models