Equi-correlated random matrices and high-dimensional statistics

概要

Equi-correlated random matrices and
high-dimensional statistics
(Summary of Doctor’s Thesis)
B9SD1701
Atina Husnaqilati
In a wide array of disciplines, the high-dimensional dataset is being generated. However, it has long been known that when the data dimension p is greater than the sample
size n, a number of well-known multivariate analysis techniques become ineffective or
even inaccurate [10, p. 1]. In such a situation, the limiting regime in random matrix
theory is considered where n, p → ∞ and p/n → c > 0. In this asymptotic framework,
Bai-Silverstein [2] and Jiang [6] studied the limiting spectral distributions (LSDs) of a
sample covariance matrix and a sample correlation matrix; a sample correlation matrix
R is important in econometrics and finance because it is invariant under the dilation and
the shifting of data.
High-dimensional datasets of econometrics and finance are often generated from factor
models where the variables are nonsparsely correlated. Among such models, fundamental
are those with a constant correlation r > 0 among variables. These models (equi-correlated
normal populations) have a population correlation matrix of order p given by
Cp (r) = (1 − r)I + rJ (0 ≤ r < 1),
where I denotes the identity matrix of order p and J is the matrix of 1 of order p.
A data matrix X = [xij ]p×n drawn from a centered population with population covariance matrix Cp (r), has a well-known decomposition
√
√
(1)
xij = rηj + 1 − rξij
where ηj and ξij (1 ≤ i ≤ p, 1 ≤ j ≤ n) are independent, standard normal random
variables. For univariate statistics, i.e., p = 1, Walsh [8] considered (1) to prove that
hypothesis tests of the univariate statistics become worse nonasymptotically when the
variables of X are mutually correlated with a positive constant r.
For a real symmetric matrix M of order p, the emprical spectral distribution (ESD)
of M is, by definition, a function F M (x) = p1 # { 1 ≤ i ≤ p | λi ≤ x } where λ1 ≥ λ2 ≥
· · · ≥ λp are the eigenvalues of M. Here, by the LSD, we mean the limit of ESD in
n, p → ∞, p/n → c > 0. By using decomposition (1), we prove the following theorem.
Theorem 1 ( [1]). Let R be a sample correlation matrix formed from a p-dimensional
normal population such that x1j , . . . , xpj are mutually correlated by r (0 ≤ r < 1) for
every j = 1, . . . , n. Suppose n, p → ∞, p/n → c > 0. Then, almost surely, the empirical
spectral distribution F R (x) converges weakly to


x
Fc
.
1−r
Here, Fc (x) is the Marčenko-Pastur distribution [10, p. 10] of index c.
Theorem 1 answers a question from Fan-Jiang [4, Remark 2.5] about the impacts of
equi-correlation coefficient r on the LSD of R. Then, among the p eigenvalues of R, for
the number k of eigenvalues of R greater than the average of the p eigenvalues, it is proved
that
(
1
(r = 0);
k
= 2
lim p,n→∞
lim
c→0
p
0 (r > 0).
p/n→c
1

This phase transition of the limit theorem elucidates mathematically rule of thumb in
high-dimensional statistics, that is, the deterioration of Guttman-Kaiser criterion [7] by
constant positive correlation among variables. The convergence of k/p to 1/2 for r = 0 is
suggested by a simulation study of Yeomans-Golder [11].
Furthermore, we present the simple methodology to show that the LSD of the product of the sample covariance matrices and the inverse of independent sample covariance
matrices (Fisher matrix [10]) from two independent equi-correlated normal populations
with correlation coefficient r1 , r2 ∈ [0, 1) is a deterministic distribution function scaled
by (1 − r2 )/(1 − r1 ) [5]. This result is a counterpart of the result from Walsh [8] about
the Snedecor F-statistics for univariate statistical analysis. Furthermore, for a matrix
Z(. . .) of linear combinations of several random variables, and for standard random variables x
ei (i = 1, 2, . . .) mutually correlated with r (0 ≤ r < 1), we consider the following
decomposition:
√
√
(2)
Z(e
x1 , x
e2 , . . .) = 1 − rZ(x1 , x2 , . . .) + rx0 Z(1, 1, . . .)
where xi (i = 0, 1, 2, . . .) are indpendent standard normal random variables. By (1), (2),
and rank inequalities for ESDs [2, p. 503], for 1 ≤ i, j ≤ p, we prove
 that the LSDs
e|i−j|+1 [3], and
of Wigner matrices p−1/2 [e
xij ] [9], symmetric Toeplitz matrices p−1/2 x
−1/2
Hankel matrices p
[e
xi+j−1 ] [3] are the same with the LSDs
of Wigner matrices ((1 −

1/2
1/2
x|i−j|+1 , and Hankel matrices
r)/p) [xij ], symmetric Toeplitz matrices ((1 − r)/p)
1/2
((1 − r)/p) [xi+j−1 ], resp.

References
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