# Multinomial Goodness-of-Fit Based on U-Statistics: High-Dimensional Asymptotic and Minimax Optimality

arXiv: https://arxiv.org/abs/1812.08924 · HTML: https://mlsanity.com/paper/1812.08924/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Dec 2018.

## Built on by (most cited first, 1 of 1)

- [Token-Efficient Change Detection in LLM APIs](https://mlsanity.com/paper/2602.11083/) (2026)

## Abstract (arXiv)

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-statistic 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.