STAT3017/STAT7017 Assignment 4 Page 1 of 2
Assignment 4
Due by Tuesday 5 October 2021 09:00
We shall consider the recent paper [A] that proposes a general framework for testing correlation
structures. That is, if we have a p-dimensional independent sample 합1, . . . ,합n and define the
sample covariance matrix
핊n = (n − 1)−1
n∑
i=1
(합i − 합¯)(합i − 합¯)T ,
the sample correlation by ℝ̂n := [diag(핊n)]−1/2핊n[diag(핊n)]−1/2, and the (true) population
correlation by ℝ. Then, for example, the one sample testing problem is concerned with testing
H0 : ℝ = ℝ∗ versus Ha : ℝ 6= ℝ∗ where ℝ∗ is a specific correlation matrix (e.g., ℝ∗ = Ip the
identity matrix).
Question 1 [10 marks]
Let us first consider the classic case where p is fixed. The asymptotic distribution of the
test statistic
핋 := (n − 1){log(|ℝ∗|/|ℝ̂n|)− p + tr(ℝ−1∗ ℝ̂n)}
often plays a role in accepting or rejecting H0.
(a)[5] Before 1969, it was (incorrectly) thought that when n →∞ the test statistic 핋 is
asymptotically distributed as χ21
2
p(p−1). Take p = 2 so that we can write ℝ and ℝ̂n
in terms of the correlation coefficient ρ and sample correlation coefficient ρˆ as
ℝ =
(
1 ρ
ρ 1
)
, ℝ̂n =
(
1 ρˆ
ρˆ 1
)
, −1 ≤ ρ, ρˆ ≤ 1.
Then using the delta method (see Week 7 and results therein), show that the correct
limiting distribution for 핋 is (1 + ρ2)χ21; see [B].
(b)[5] Perform a simulation study to verify your result in (a) for p = 2 and various choices of
ρ. Do this by repeatedly sampling sample covariances from the Wishart distribution,
transforming these matrices to correlation matrices, and calculating the test statistic.
Show a figure with the histogram of the test statistic against the appropriately scaled
and parametrised χ2 density overlayed on top.
Question 2 [10 marks]
For the high-dimensional case p, n → ∞ such that p/n → y ∈ (0,∞) , it is proposed
in [A] to consider the test statistic
핃 := tr[(ℝ̂n −ℝ∗)2].
Dale Roberts - Australian National University
Last updated: June 7, 2021
STAT3017/STAT7017 Assignment 4 Page 2 of 2
(a)[4] In Theorem 2.1, it is shown that asymptotically
(핃− µz0)/(2(n − 1)−1 tr(ℝ2∗)) d→ N(0, 1).
Perform a simulation study to show this in the cases (i) ℝ∗ = Ip, and (ii) ℝ∗ =
(ρ|i−j |)pi,j=1 for ρ = 0.5. Do this for p = 50, 100, 200, 500. You will need to look at
the paper to determine the correct choice of µz0.
(b)[6] Implement the full test given by Eq. (2.1) and (2.2) in the paper and plot the empirical
sizes of the method for p = 50, 100, 200, 500 in the case where ℝ∗ = (ρ|i−j |)pi,j=1 for
ρ = 0.5.
References
[A] Zheng, Cheng, Guo, and Zhu (2019). Test for high-dimensional correlation matrices. Annals of Statistics,
Vol. 47, No. 5.
[B] Aitkin (1969). Some tests for correlation matrices. Biometrika, Vol. 56, No. 2.
Note: I have placed these references in the ‘Readings’ folder on Wattle.
Dale Roberts - Australian National University
Last updated: June 7, 2021

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