Multinomial goodness-of-fit based on U-statistics: High-dimensional asymptotic and minimax optimality

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Abstract

We consider multinomial goodness-of-fit tests in the high-dimensional regime where the number of bins increases with the sample size. In this regime, Pearson's chi-squared test can suffer from low power due to the substantial bias as well as high variance of its statistic. To resolve these issues, we introduce a family of U-statistics for multinomial goodness-of-fit and study their asymptotic behaviors in high-dimensions. Specifically, we establish conditions under which the considered U-statistic is asymptotically Poisson or Gaussian, and investigate its power function under each asymptotic regime. Furthermore, we introduce a class of weights for the U-statistic that results in minimax rate optimal tests.

Original languageEnglish
Pages (from-to)74-91
Number of pages18
JournalJournal of Statistical Planning and Inference
Volume205
DOIs
Publication statusPublished - 2020 Mar

Bibliographical note

Publisher Copyright:
© 2019 Elsevier B.V.

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

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