Hypothesis Testing for High-Dimensional Multinomials: A Selective Review

By Sivaraman Balakrishnan, Larry A. Wasserman. 2017

68 citations recorded by the source corpus.

The statistical analysis of discrete data has been the subject of extensive statistical research dating back to the work of Pearson. In this survey we review some recently developed methods for testing hypotheses about high-dimensional multinomials. Traditional tests like the $\chi^2$ test and the likelihood ratio test can have poor power in the high-dimensional setting. Much of the research in this area has focused on finding tests with asymptotically Normal limits and developing (stringent) conditions under which tests have Normal limits. We argue that this perspective suffers from a significant deficiency: it can exclude many high-dimensional cases when - despite having non Normal null distributions - carefully designed tests can have high power. Finally, we illustrate that taking a minimax perspective and considering refinements of this perspective can lead naturally to powerful and practical tests.

Subfields: Artificial Intelligence, Management Science and Operations Research, Statistics and Probability

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Citations in the corpus

Inference Under Information Constraints I: Lower Bounds From Chi-Square Contraction (2018, background): “d equal to Θ( √ k/ε2). As mentioned earlier, in this work we are following this samplecomplexityframework that has received attention in recent years. We refer the reader to surveys [17], [41], [14], [7] for a comprehensive review of recent results on discrete distribution learning and testing. Dist”
Inference Under Information Constraints II: Communication Constraints and Shared Randomness (2019, background): “However, the sample-optimal setting of interest to us has received a lot of attention in the past decade, especially in the computer science literature; see [42], [18], [9] for surveys.”
Minimax optimality of permutation tests (2020, background): “However, as noted in Balakrishnan and Wasserman (2018), this can exclude many high-dimensional cases when – despite having non-normal null distributions – carefully designed tests can have high (minimax) power.”
Test without Trust: Optimal Locally Private Distribution Testing (2018, background): “This literature itself spans the last two decades; we refer an interested reader to surveys and books [30, 11, 24, 8] for a comprehensive review.”
Locally private non-asymptotic testing of discrete distributions is faster using interactive mechanisms (2020, background): “It has been known since Valiant and Valiant [2014], see also Balakrishnan and Wasserman [2018], that in the direct setup a weighted `2-test is needed in order to attain the optimal rates.”
Unifying Human and Statistical Evaluation for Natural Language Generation (2019, background): “It is known that total variation distance estimates have pessimistic minimax estimation rates in high dimensions (Balakrishnan and Wasserman, 2017).”
Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates (2017, background): “Finally, it would also be interesting to further explore the extent to which the local-minimax perspective can lead to a better understanding of composite-null inference problems [1, 2, 5, 8, 10, 23, 30].”
Inference Under Information Constraints III: Local Privacy Constraints (2021, background): “In this section, we only focus on closely related papers and we refer an interested reader to surveys and books [13], [21], [34], [43] for a comprehensive treatment.”
Minimax optimal conditional independence testing (2020, methodology): “This is in sharp contrast to typical tests for goodness-of-fit and twosample tests (for instance those analyzed in [2, 4, 5, 12, 35]).”