Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions

Let [Formula: see text] be a [Formula: see text]-dimensional normal distribution. Testing [Formula: see text] equal to a given matrix or [Formula: see text] equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension [Formula: see text] is fixed, it is known that the LRT statistics go to [Formula: see text]-distributions. When [Formula: see text] is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that [Formula: see text] and [Formula: see text] are proportional to each other. The condition [Formula: see text] suffices in our results.

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Moderate deviation principles for classical likelihood ratio tests of high-dimensional normal distributions

Journal of Multivariate Analysis ◽

10.1016/j.jmva.2017.02.004 ◽

2017 ◽

Vol 156 ◽

pp. 57-69 ◽

Cited By ~ 4

Author(s):

Hui Jiang ◽

Shaochen Wang

Keyword(s):

Likelihood Ratio ◽

Moderate Deviation ◽

Likelihood Ratio Tests ◽

High Dimensional ◽

Normal Distributions

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Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles

Japanese Journal of Statistics and Data Science ◽

10.1007/s42081-020-00096-7 ◽

2020 ◽

Author(s):

Yuta Koike

Keyword(s):

Central Limit ◽

Limit Theorems ◽

Central Limit Theorems ◽

High Dimensional

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Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

Scandinavian Journal of Statistics ◽

10.1111/sjos.12383 ◽

2019 ◽

Vol 46 (2) ◽

pp. 636-660 ◽

Cited By ~ 2

Author(s):

Taras Bodnar ◽

Stepan Mazur ◽

Nestor Parolya

Keyword(s):

Covariance Matrix ◽

Central Limit ◽

Limit Theorems ◽

Central Limit Theorems ◽

Sample Covariance Matrix ◽

Normal Distributions ◽

Large Sample ◽

Sample Covariance ◽

Mixture Of Normal Distributions ◽

Mean Vector

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High-dimensional limit theorems for random vectors in ℓpn-balls

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500924 ◽

2019 ◽

Vol 21 (01) ◽

pp. 1750092 ◽

Cited By ~ 7

Author(s):

Zakhar Kabluchko ◽

Joscha Prochno ◽

Christoph Thäle

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Limit Theorems ◽

Large Deviation ◽

Rate Function ◽

Central Limit Theorems ◽

High Dimensional ◽

Random Vectors ◽

Random Direction

In this paper, we prove a multivariate central limit theorem for [Formula: see text]-norms of high-dimensional random vectors that are chosen uniformly at random in an [Formula: see text]-ball. As a consequence, we provide several applications on the intersections of [Formula: see text]-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an [Formula: see text]-ball onto a line spanned by a random direction [Formula: see text]. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime [Formula: see text] this displays in speed and rate function deviations of the [Formula: see text]-norm on an [Formula: see text]-ball obtained by Schechtman and Zinn, but we obtain explicit constants.

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