Likelihood Ratio Tests for High‐Dimensional Normal Distributions

By Tiefeng Jiang, Fan Yang. 2013

149 citations recorded by the source corpus.

In their recent work, Jiang and Yang studied six classical Likelihood Ratio Test statistics under high‐dimensional setting. Assuming that a random sample of size n is observed from a p‐dimensional normal population, they derive the central limit theorems (CLTs) when p and n are proportional to each other, which are different from the classical chi‐square limits as n goes to infinity, while p remains fixed. In this paper, by developing a new tool, we prove that the mentioned six CLTs hold in a more applicable setting: p goes to infinity, and p can be very close to n. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi‐square approximations and discussions are presented afterwards.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/61ab84984b36cd9e9609378f142d75e3ca85e229

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Mathias Drton
2. Fang Han
3. Hongjian Shi
4. Tiefeng Jiang
5. Aaditya Ramdas
6. Jiang Hu
7. Danning Li
8. Guangming Pan

In Statistics and Probability

1. Mathias Drton
2. Fang Han
3. Hongjian Shi
4. Wang Zhou
5. Guangming Pan
6. Tiefeng Jiang
7. Z. Bao
8. Aaditya Ramdas

Citations in the corpus

• Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case (2017, background): “Currently, for high dimensional cases, there are three kinds of widely discussed test procedures: (i) Corrected likelihood ratio tests (see [29, 30]); (ii) Trace tests (see [29, 49, 28, 47, 8]); (iii) Largest eigenvalue tests (see [32, 33, 26]).”
• Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations (2013, background): “Jiang and Yang (2013) and Jiang and Qi (2013) investigate the test through obtaining the central limit theorem of its likelihood ratio test statistic under p/n→ y ∈ [0, 1].”
• On testing the equality of high dimensional mean vectors with unequal covariance matrices (2014, background): “at the commonly used likelihood ratio test performs badly when dimension is large has been considered in a lot of literature such as Bai and Saranadasa (1996); Bai et al. (2009); Jiang et al. (2012); Jiang and Yang (2013), the discussion of the likelihood ratio test is left out in this paper. Recent”
• False discovery rate control with e‐values (2020, background): “There are several results on the high-dimensional asymptotics of likelihood ratios, but the resulting p-value relies on the practitioner making assumptions on the relative scalings of n and d; see Jiang and Yang (2013) for example.”
• High-dimensional consistent independence testing with maxima of rank correlations (2018, background): “For example, Bai et al. (2009) and Jiang and Yang (2013) derived corrected likelihood ratio tests for Gaussian data.”