The paper considers the likelihood ratio (LR) test on the block circular symmetric covariance structure of a multivariate Gaussian population with divergent dimension. When the sample size n, the dimension of each block p and the number of blocks u satisfy pu < n − 1 and p = p(n) → ∞ as n → ∞, the asymptotic distribution and the moderate deviation principle of the logarithmic LR test statistic under the null hypothesis are established. Some numerical simulations indicate that the proposed LR test method performs well in the divergent-dimensional block circular symmetric covariance structure test.
Accepté le :
DOI : 10.1051/ps/2019020
Mots-clés : Likelihood ratio test, block circular symmetric model, asymptotic normality, moderate deviation principle
@article{PS_2019__23__672_0,
author = {Xie, Junshan and Sun, Gaoming},
title = {A test for block circular symmetric covariance structure with divergent dimension},
journal = {ESAIM: Probability and Statistics},
pages = {672--696},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2019020},
mrnumber = {4011570},
zbl = {1507.62255},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2019020/}
}
TY - JOUR AU - Xie, Junshan AU - Sun, Gaoming TI - A test for block circular symmetric covariance structure with divergent dimension JO - ESAIM: Probability and Statistics PY - 2019 SP - 672 EP - 696 VL - 23 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2019020/ DO - 10.1051/ps/2019020 LA - en ID - PS_2019__23__672_0 ER -
%0 Journal Article %A Xie, Junshan %A Sun, Gaoming %T A test for block circular symmetric covariance structure with divergent dimension %J ESAIM: Probability and Statistics %D 2019 %P 672-696 %V 23 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2019020/ %R 10.1051/ps/2019020 %G en %F PS_2019__23__672_0
Xie, Junshan; Sun, Gaoming. A test for block circular symmetric covariance structure with divergent dimension. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 672-696. doi : 10.1051/ps/2019020. http://archive.numdam.org/articles/10.1051/ps/2019020/
[1] , An Introduction to Multivariate Statistical Analysis. John Wiley & Sons Inc., New York (1984). | MR | Zbl
[2] , Application of the theory of products of problems to certain patterned matrics. Ann. Stat. 1 (1973) 682–699. | DOI | MR | Zbl
[3] and , Effect of high dimension: by an example of a two sample problem. Stat. Sin. 6 (1996) 311–329. | MR | Zbl
[4] , Applied Matrix Algebra in the Statistical Sciences. North-Holland, New York (1983). | MR | Zbl
[5] , , and , A maximum entropy solution of the covariance extension problem for reciprocal processes. IEEE Trans. Autom. Control, 56 (2011) 1999–2012. | DOI | MR | Zbl
[6] , , and , An efficient algorithm for maximum-entropy extension of block-circulant covariance matrices. Linear Algebra Appl. 439 (2011) 2309–2329. | DOI | MR | Zbl
[7] and , Probability Theory: Independence, Interchangeability, Martingale, 3rd edn. Springer-Verlag, New York (1997). | DOI | MR | Zbl
[8] , The eigenblock and eigenmatrix decomposition of a matrix: its usefulness in statistics-application to the likelihood ratio test for block-circularity. (2013), Preprint.
[9] and , Large Deviations Techniques and Applications. Springer, New York (2009). | MR | Zbl
[10] and , A spatial analysis of variance applied to soil-water infiltration. Water Resour. Res. 26 (1990) 2695–2703. | DOI
[11] , Probability: A Graduate Course. Springer-Verlag, New York (2005). | MR | Zbl
[12] and , Estimation of familial correlations under autoregressive circular covariance. Commun. Stat. Theory Methods 30 (2001) 1811–1828. | DOI | MR | Zbl
[13] and , Moderate deviation principles for classical likelihood ratio tests of high-dimensional normal distributions. J. Multivar. Anal. 156 (2017) 57–69. | DOI | MR | Zbl
[14] and , Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Ann. Stat. 41 (2013) 2029–2074. | DOI | MR | Zbl
[15] and , Likelihood ratio tests for high-dimensional normal distributions. Scand. J. Stat. 42 (2015) 988–1009. | DOI | MR | Zbl
[16] , and , High-dimensional asymptotic expansion of LR statistic for testing intraclass correlation structure and its error bound. J. Multivar. Anal. 101 (2010) 101–112. | DOI | MR | Zbl
[17] , Linear discrimination with equicorrelated training vectors. J. Multivar. Anal. 98 (2007) 384–409. | DOI | MR | Zbl
[18] , and , Block circular symmetry in multilevel models, in Research Report 2011, vol. 3, Department of Statistics, Stockholm University (2011).
[19] , and , On estimation in multilevel models with block circular symmetric covariance structures. Acta et Commentationes Universitatis Tartuensis de Mathematica 16 (2012) 1–14. | DOI | MR | Zbl
[20] , and , On estimation in hierarchical models with block circular covariance structures. Ann. Inst. Stat. Math. 67 (2015) 1–19. | DOI | MR | Zbl
[21] and , Modeling of stationary periodic time series by ARMA representations, in Optimization and Its Applications in Control and Data Sciences, edited by . Springer Optimization and Its Applications, Vol 115. Springer, Cham (2016). | DOI | MR
[22] , and , Likelihood ratio test statistic for block compound symmetry covariance structure and its asymptoic expansion. Technical Report No.15-03, Statistical Research Group, Hiroshima University, Japan (2015).
[23] and , Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions. Comput. Stat. 28 (2013) 2091–2115. | DOI | MR | Zbl
[24] , Aspects of Multivariate Statistical Theory. John Wiley & Sons Inc., New York (1982). | DOI | MR | Zbl
[25] , and , Distribution and percentage points of the likelihood ratio statistic for testing circular symmetry. Comput. Stat. Data Anal. 47 (2004) 79–89. | DOI | MR | Zbl
[26] , and , On testing circular stationarity and related models. J. Stat. Comput. Simul. 29 (1988) 225–239. | DOI | MR | Zbl
[27] and , Shift permutation invariance in linear random factor models. Math. Methods Stat. 17 (2008) 173–185. | DOI | MR | Zbl
[28] , Testing and estimation for structures which are circularly symmetric in blocks, in Multivariate Statistical Inference, edited by , . North-Holland, Amsterdam (1973) 183–195. | MR | Zbl
[29] and , Testing and estimation for a circular stationary model. Ann. Math. Stat. 40 (1969) 1358–1373. | DOI | MR | Zbl
[30] , Familial correlations or the multivariate generalizations of the intraclass correlation. Curr. Sci. 14 (1945) 66–67.
[31] , Discriminant functions for genetic differentiation and selection. Sankhya 12 (1953) 229–246. | MR | Zbl
[32] and , Estimating and testing a structured covariance matrix for three-level multivariate data. Commun. Stat. Theory Methods 40 (2011) 1945–1963. | DOI | MR | Zbl
[33] , Estimation of intraclass correlations in familial data. Biometrika 71 (1984) 177–185. | DOI | MR | Zbl
[34] , Sample criteria for testing equality of means, equality of variances, and equality of covariances in a normal multivariate distribution. Ann. Math. Stat. 17 (1946) 257–281. | DOI | MR | Zbl
[35] and , A high-dimensional likelihood ratio test for circular symmetric covariance structure. Commun. Stat. Theory Methods 47 (2018) 1392–1402. | DOI | MR | Zbl
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