The Annals of Statistics

Tests for covariance structures with high-dimensional repeated measurements

Ping-Shou Zhong, Wei Lan, Peter X. K. Song, and Chih-Ling Tsai

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In regression analysis with repeated measurements, such as longitudinal data and panel data, structured covariance matrices characterized by a small number of parameters have been widely used and play an important role in parameter estimation and statistical inference. To assess the adequacy of a specified covariance structure, one often adopts the classical likelihood-ratio test when the dimension of the repeated measurements ($p$) is smaller than the sample size ($n$). However, this assessment becomes quite challenging when $p$ is bigger than $n$, since the classical likelihood-ratio test is no longer applicable. This paper proposes an adjusted goodness-of-fit test to examine a broad range of covariance structures under the scenario of “large $p$, small $n$.” Analytical examples are presented to illustrate the effectiveness of the adjustment. In addition, large sample properties of the proposed test are established. Moreover, simulation studies and a real data example are provided to demonstrate the finite sample performance and the practical utility of the test.

Article information

Ann. Statist., Volume 45, Number 3 (2017), 1185-1213.

Received: May 2015
Revised: March 2016
First available in Project Euclid: 13 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing
Secondary: 62G20: Asymptotic properties 62G10: Hypothesis testing

Adjusted test goodness-of-fit test longitudinal data panel data


Zhong, Ping-Shou; Lan, Wei; Song, Peter X. K.; Tsai, Chih-Ling. Tests for covariance structures with high-dimensional repeated measurements. Ann. Statist. 45 (2017), no. 3, 1185--1213. doi:10.1214/16-AOS1481.

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Supplemental materials

  • Supplement to “Tests for covariance structures with high-dimensional repeated measurements”. This supplemental material provides technical proofs of the main results in Section 3, some asymptotic results on the proposed test statistics for nonnormally distributed random vectors, and the proof of the asymptotic normality of the test statistic given in Section 6. We also provide the proof of Remark 4 and present additional numerical simulation experiments to compare our proposed test statistics with some existing methods.