# Sure Screening for Transelliptical Graphical Models

Yuxiang Xie, Chengchun Shi, Rui Song

(January 15, 2016)

###### Abstract

We propose a sure screening approach for recovering the structure of a
transelliptical graphical model in the high dimensional setting. We estimate
the partial correlation graph by thresholding the elements of an estimator of
the sample correlation matrix obtained using Kendall’s tau statistic. Under a
simple assumption on the relationship between the correlation and partial
correlation graphs, we show that with high probability, the estimated edge set
contains the true edge set, and the size of the estimated edge set is
controlled. We develop a threshold value that allows for control of the
expected false positive rate. In simulation and on an equities data set, we
show that transelliptical graphical sure screening performs quite
competitively with more computationally demanding techniques for graph
estimation.
some key words: High Dimensionality; Kendall’s tau; Partial Correlation;
Sparsity; Undirected Graph

##  1 Introduction

Consider the random vector $X=\left(X_{1},\ldots,X_{p}\right)^{T}$, and an
undirected graph denoted by $G=\left(\mathcal{V},\mathcal{E}\right)$, where
$\mathcal{V}=\left\\{1,\dots,p\right\\}$ is the set of nodes, and
$\mathcal{E}$ is the set of edges describing the conditional dependence
relationships among $X=\left(X_{1},\ldots,X_{p}\right)^{T}$. A pair
$\left(j,j^{\prime}\right)$ is contained in the edge set $\mathcal{E}$ if and
only if $X_{j}$ is conditionally dependent on $X_{j^{\prime}}$, given all
remaining variables
$X_{\mathcal{V}\backslash\\{j,j^{\prime}\\}}=\left\\{X_{k};k\in\mathcal{V}\backslash\left\\{j,j^{\prime}\right\\}\right\\}$.

In recent years, many methods have been developed to recover the structure of
graphical models in the high dimensional setting. Many authors have studied
the Gaussian graphical model, in which conditional dependence is encoded by
the sparsity pattern of the inverse covariance matrix (Yuan and Lin,, 2007;
Friedman et al.,, 2008; Rothman et al.,, 2008; Ravikumar et al.,, 2009, and
references therein). Liu et al., (2009); Liu et al., 2012a  introduced the
nonparanormal distribution, which results from univariate monotonic
transformations of the Gaussian distribution, and showed that the structural
properties of the inverse covariance matrix of the Gaussian distribution carry
over to the corresponding nonparanormal distribution. Ravikumar et al., (2010)
and Anandkumar et al., (2012) considered recovering the structure of an Ising
graphical model. Yang et al., (2012) studied a class of graphical models in
which the node-wise conditional distributions arise from the exponential
family. Moreover, Yang et al., (2014) and Chen et al., (2015) considered the
problem of structure recovery for mixed graphical models.

Luo et al., (2015) proposed a computationally-efficient screening approach for
Gaussian graphical models, which they called _graphical sure screening_
(GRASS). They estimated an edge between the $j$th and $j^{\prime}$th nodes if
the sample correlation between the $j$th and $j^{\prime}$th features exceeds
some threshold $\gamma$. Then the $j$th node’s estimated neighborhood contains
the true neighborhood with very high probability, under certain simple
assumptions. GRASS requires only $\mathcal{O}(p^{2})$ operations, while most
other existing methods for estimating the graph require $\mathcal{O}(p^{3})$
computations (Friedman et al.,, 2008).

However, GRASS requires that the data follows a multivariate normal
distribution. In many settings, a reliance on exact normality is not desirable
(Liu et al.,, 2009; Liu et al., 2012a, ; Liu et al., 2012b, ). In this paper,
we propose _transelliptical GRASS_ , an extension of the GRASS procedure to
the transelliptical graphical model family, introduced by Liu et al., 2012b .
We show that under a certain set of simple assumptions, the desirable
statistical properties held by GRASS are also held by transelliptical GRASS.
However, due to the relaxation of multivariate normality, the graph that we
estimate represents the partial correlation, rather than conditional
dependence, among variables.

The rest of this paper is organized as follows. In Section 2, we provide some
background on transelliptical graphical models, and present a useful property
of Kendall’s tau statistic. In Section 3, we establish the theoretical
properties for transelliptical sure screening, which include the sure
screening property, size control of the selected edge set, and the control of
the expected false positive rate. Furthermore, we provide a choice of the
threshold value that leads to the aforementioned desirable properties.
Simulation studies are presented in Section 4, and an application to an
equities data set is shown in Section 5. We close with a discussion in Section
6.

##  2 Preliminaries

The transelliptical distribution (Liu et al., 2012b, ) is a generalization of
the nonparanormal distribution (Liu et al.,, 2009; Liu et al., 2012a, ). The
transelliptical distribution extends the elliptical distribution in much the
same way that the nonparanormal extends the normal distribution. We first
provide the definition of the elliptical distribution.

######  Definition 1.

Let $\mu\in\mathbb{R}^{p}$ and $\Sigma\in\mathbb{R}^{p\times p}$ with
$\mathrm{rank}\left(\Sigma\right)=q\leq p$. A $p$-dimensional random vector
$X$ has an elliptical distribution, denoted by
$X\sim\mbox{EC}_{p}\left(\mu,\Sigma,r\right)$, if it has a stochastic
representation

$\displaystyle X\stackrel{{\scriptstyle d}}{{=}}\mu+rAU,$ (2.1)

where $U$ is a random vector uniformly distributed on the unit sphere in
$\mathbb{R}^{q}$, $r\geq 0$ is a scalar random variable independent of $U$,
and $A\in\mathbb{R}^{p\times q}$ satisfies $\Sigma=AA^{T}$.

In Definition 1, $X\stackrel{{\scriptstyle d}}{{=}}Y$ indicates that $X$ and
$Y$ have the same distribution. Many multivariate distributions, such as the
multivariate normal and the multivariate t-distribution, belong to the
elliptical distribution family.

From now on, we assume that $A$ in (2.1) is a full rank $p\times p$ matrix so
that $\Sigma$ is positive definite. In addition, assume that $\Sigma$ has unit
diagonal elements. We let $\rho_{jj^{\prime}}$ denote the
$\left(j,j^{\prime}\right)$th entry of $\Sigma$. We also assume that the
scalar random variable $r$ in (2.1) has density $g(\cdot)$ and
${\mathbb{E}}(r^{2})<\infty$.

######  Definition 2.

A continuous random vector $X=\left(X_{1},\ldots,X_{p}\right)^{T}$ follows a
$p$-dimensional transelliptical distribution, denoted by
$\mbox{TE}_{p}\left(\Sigma,r;f_{1},\dots,f_{p}\right)$, if there exist
monotone univariate functions $\left\\{f_{j}\right\\}_{j=1}^{p}$ and a
nonnegative random variable $r$ satisfying $\Pr(r=0)=0$, such that $Z\equiv
f(X)\equiv\left(f_{1}\left(X_{1}\right),\ldots,f_{p}\left(X_{p}\right)\right)^{T}\sim\mbox{EC}_{p}\left(0,\Sigma,r\right)$.
We refer to
$Z=\left(Z_{1},\ldots,Z_{p}\right)^{T}=\left(f_{1}\left(X_{1}\right),\ldots,f_{p}\left(X_{p}\right)\right)^{T}$
as the latent variables of $X=\left(X_{1},\ldots,X_{p}\right)^{T}$.

######  Remark 1.

A random vector $X=\left(X_{1},\ldots,X_{p}\right)^{T}$ follows a
nonparanormal distribution (Liu et al.,, 2009; Liu et al., 2012a, ) if there
exist monotone univariate functions $\left\\{f_{j}\right\\}_{j=1}^{p}$ such
that
$Z=f(X)=\left(f_{1}\left(X_{1}\right),\ldots,f_{p}\left(X_{p}\right)\right)^{T}\sim
N_{p}({0},{\Sigma})$. Therefore, the transelliptical distribution is a strict
extension of the nonparanormal distribution.

Given a transelliptical distribution
$\mbox{TE}_{p}\left(\Sigma,r;f_{1},\dots,f_{p}\right)$, we can define an
undirected graph $G=\left(\mathcal{V},\mathcal{E}\right)$, where
$\mathcal{V}=\left\\{1,\ldots,p\right\\}$, and
$\left(j,j^{\prime}\right)\in\mathcal{E}$ if and only if
$\left(\Sigma^{-1}\right)_{jj^{\prime}}\neq 0$ — that is, if and only if the
latent variables $Z_{j}=f_{j}(X_{j})$ and
$Z_{j^{\prime}}=f_{j^{\prime}}(X_{j^{\prime}})$ are partially correlated (Liu
et al., 2012b, ). In the special case of a nonparanormal distribution, a zero
entry of the precision matrix $\Sigma^{-1}$ further implies conditional
independence between the corresponding pair of random variables.

Next, we present the definition and some theoretical properties of the
Kendall’s tau statistic.

######  Definition 3.

Given $n$ independent draws from $X=(X_{1},\ldots,X_{p})^{T}$, such that
$X_{ij}$ is the value of the $j$th variable in the $i$th observation, the
population-level Kendall’s tau statistic between $X_{j}$ and $X_{j^{\prime}}$
is defined as

$\tau_{jj^{\prime}}={\mathbb{E}}\left[\mathrm{sign}\left(\left(X_{1j}-X_{2j}\right)\left(X_{1j^{\prime}}-X_{2j^{\prime}}\right)\right)\right].$
(2.2)

The sample estimator of Kendall’s tau is defined as

$\widehat{\tau}_{jj^{\prime}}=\frac{2}{n(n-1)}\sum_{1\leq i\leq i^{\prime}\leq
n}\mathrm{sign}\left(\left(X_{ij}-X_{i^{\prime}j}\right)\left(X_{ij^{\prime}}-X_{i^{\prime}j^{\prime}}\right)\right).$
(2.3)

Now suppose that we have $n$ independent draws from
$X=(X_{1},\ldots,X_{p})^{T}\sim\mbox{TE}_{p}(\Sigma,r;f_{1},\dots,f_{p})$,
such that $X_{ij}$ is the value of the $j$th variable in the $i$th
observation. Then there is a simple connection between the population-level
Kendall’s tau statistic $\tau_{jj^{\prime}}$ and $\rho_{jj^{\prime}}$.

######  Lemma 1.

(Liu et al., 2012b, ) For
$X=\left(X_{1},\ldots,X_{p}\right)^{T}\sim\mbox{TE}_{p}(\Sigma,r;f_{1},\dots,f_{p})$,
we have $\rho_{jj^{\prime}}=\sin(\frac{\pi}{2}\tau_{jj^{\prime}})$.

Lemma 1 motivated Liu et al., 2012b  to estimate $\rho_{jj^{\prime}}$ using

$\widehat{S}^{\tau}_{jj^{\prime}}=\begin{cases}\sin\left(\frac{\pi}{2}\widehat{\tau}_{jj^{\prime}}\right)&\text{if
$j\neq j^{\prime}$}\\\ 1&\text{if $j=j^{\prime}$}\end{cases}.$ (2.4)

##  3 Transelliptical Graphical Sure Screening

###  3.1 Proposed Approach

Suppose that we have $n$ independent draws from
$X=(X_{1},\ldots,X_{p})^{T}\sim\mbox{TE}_{p}(\Sigma,r;f_{1},\dots,f_{p})$. We
define
$\mathcal{E}\equiv\left\\{\left(j,j^{\prime}\right):j<j^{\prime},\left(\Sigma^{-1}\right)_{jj^{\prime}}\neq
0\right\\}$ to be the true edge set, and
$\mathcal{E}_{j}\equiv\left\\{j^{\prime}:j^{\prime}\neq
j,\left(\Sigma^{-1}\right)_{jj^{\prime}}\neq 0\right\\}$ to be the true
neighborhood for the $j$th node. We propose to estimate $\mathcal{E}$ and
$\mathcal{E}_{j}$ as follows,

$\widehat{\mathcal{E}}_{\gamma_{n}}=\left\\{\left(j,j^{\prime}\right):j<j^{\prime},\left|\widehat{S}^{\tau}_{jj^{\prime}}\right|>\gamma_{n,jj^{\prime}}\right\\}$
(3.1)

and

$\widehat{\mathcal{E}}_{j,\gamma_{n}}=\left\\{j^{\prime}:j^{\prime}\neq
j,\left|\widehat{S}^{\tau}_{jj^{\prime}}\right|>\gamma_{n,jj^{\prime}}\right\\},$
(3.2)

where $\gamma_{n,jj^{\prime}}>0$ is some threshold value that we will specify
in the following sections, and $\widehat{S}^{\tau}_{jj^{\prime}}$ is defined
in (2.4). We refer to $\widehat{\mathcal{E}}_{\gamma_{n}}$ and
$\widehat{\mathcal{E}}_{j,\gamma_{n}}$ as the _transelliptical graphical sure
screening_ (transelliptical GRASS) estimators.

###  3.2 Theoretical Properties

We now present some theoretical properties of transelliptical GRASS. Proofs
are in the Appendix.

######  Assumption 1.

For some constant $C_{1}>0$ and $0<\kappa<1/2$,

$\min_{\left(j,j^{\prime}\right)\in\mathcal{E}}\left|\rho_{jj^{\prime}}\right|\geq
C_{1}n^{-\kappa}.$

Assumption 1 requires that the elements in the edge set $\mathcal{E}$
correspond to sufficiently large values in the correlation matrix. We next
present the sure screening property in Theorem 1.

######  Theorem 1.

Suppose that Assumption 1 holds, and that $\log(p)=C_{3}n^{\xi}$ for some
constants $C_{3}>0$ and $\xi\in(0,1-2\kappa)$. Let
$\gamma_{n,jj^{\prime}}\equiv\gamma_{n}=\frac{2}{3}C_{1}n^{-\kappa}$. Then
there exist constants $C_{4}$ and $C_{5}$ such that

$\displaystyle\Pr(\mathcal{E}\subseteq\widehat{\mathcal{E}}_{\gamma_{n}})$
$\displaystyle\geq 1-C_{4}\exp(-C_{5}n^{1-2\kappa})$

and

$\displaystyle\Pr(\mathcal{E}_{j}\subseteq\widehat{\mathcal{E}}_{j,\gamma_{n}})$
$\displaystyle\geq 1-C_{4}\exp(-C_{5}n^{1-2\kappa}).$

Conversely, if
$\min_{(j,j^{\prime})\in\mathcal{E}}|\rho_{jj^{\prime}}|<C_{1}n^{-\kappa}/3$,
then there exist constants $C_{6}$ and $C_{7}$ such that

$\displaystyle\Pr(\mathcal{E}\not\subseteq\widehat{\mathcal{E}}_{\gamma_{n}}){\geq}1-C_{6}\exp(-C_{7}n^{1-2\kappa}).$
(3.3)

Theorem 1 guarantees that the candidate edge set obtained from transelliptical
GRASS contains the true edge set with high probability, which means the
screening method will not result in false negatives with high probability.
Moreover, (3.3) suggests that Assumption 1 is necessary up to a constant. The
following corollary shows that under Assumption 1, transelliptical GRASS can
recover the connected components of $\mathcal{E}$ with high probability.

######  Corollary 1.

Suppose there are $h$ connected components in the graph $\mathcal{E}$, and
that the $l$th connected component contains the variables
$x_{1}^{(l)},\dots,x_{p_{l}}^{(l)}$, where $\sum_{l=1}^{h}p_{l}=p$. That is,
$x_{j}^{(s)}$ and $x_{j^{\prime}}^{(t)}$ are partially uncorrelated for
$s\not=t$. Suppose Assumption 1 and the conditions in Theorem 1 hold. Let
$\gamma_{n,jj^{\prime}}=\gamma_{n}=2C_{1}n^{-\kappa}/3$. Then the connected
components of $\widehat{\mathcal{E}}_{\gamma_{n}}$ are the same as the
connected components of $\mathcal{E}$ with probability at least
$1-C_{4}\exp\left(-C_{5}n^{1-2\kappa}\right)$.

Our next theorem will provide a bound on the size of
$\widehat{\mathcal{E}}_{j,\gamma_{n}}$. This requires an additional
assumption.

######  Assumption 2.

There exist constants $\alpha\geq 0$ and $C_{2}>0$ such that
$\Lambda_{\max}(\Sigma)\leq C_{2}n^{\alpha}$, where $\Lambda_{\max}(\Sigma)$
is the largest eigenvalue of $\Sigma$.

Assumption 2 allows the largest eigenvalue of the population covariance matrix
$\Sigma$ to diverge as $n$ grows.

######  Theorem 2.

Let $\gamma_{n,jj^{\prime}}=\gamma_{n}=\frac{2}{3}C_{1}n^{-\kappa}$. Under
Assumptions 1–2, if $\log\left(p\right)=C_{3}n^{\xi}$ for some constants
$C_{3}>0$ and $\xi\in\left(0,1-2\kappa\right)$, then
$\Pr\left(\left|\widehat{\mathcal{E}}_{j,\gamma_{n}}\right|\leq
O\left(n^{2\kappa+\alpha}\right)\right)$$\geq
1-C_{4}\exp\left(-C_{5}n^{1-2\kappa}\right)$, where the constants $C_{4}$ and
$C_{5}$ are as in Theorem 1.

Next we propose a choice of the threshold $\gamma_{n,jj^{\prime}}$ that
enables us to control the expected false positive rate, defined as
${|\widehat{\mathcal{E}}_{\gamma_{n}}\cap\mathcal{E}^{c}|}/{\mathcal{E}^{c}}$,
at a pre-specified value. Here,
$\mathcal{E}^{c}\equiv\left\\{\left(j,j^{\prime}\right):j<j^{\prime},\left(\Sigma^{-1}\right)_{jj^{\prime}}=0\right\\}$,
and $\widehat{\mathcal{E}}_{\gamma_{n}}$ is defined in (3.1). This requires an
additional assumption.

######  Assumption 3.

For the same $\xi$ as in Theorem 1,

$\max_{(j,j^{\prime})\notin\mathcal{E}}\left|\rho_{jj^{\prime}}\right|=o\left(n^{-\frac{1-\xi}{2}}\right).$

######  Theorem 3.

Suppose that Assumptions 1–3 hold. If $\log(p)=C_{3}n^{\xi}$ for $\xi$ defined
in Theorem 1, then we can control the asymptotic expected false positive rate
at $f/|\mathcal{E}^{c}|$ by choosing
$\gamma_{n,jj^{\prime}}=\frac{\pi}{2}\widehat{\omega}_{jj^{\prime}}\Phi^{-1}(1-\frac{f}{p(p-1)})/\sqrt{n}$,
where

$\widehat{\omega}_{jj^{\prime}}^{2}=\frac{4(n-1)}{(n-2)^{2}}\sum_{i^{\prime}=1}^{n}\left[\left\\{\frac{1}{n-1}\sum_{i=1,i\neq
i^{\prime}}^{n}\mathrm{sign}\left(\left(X_{ij}-X_{i^{\prime}j}\right)\left(X_{ij^{\prime}}-X_{i^{\prime}j^{\prime}}\right)\right)\right\\}-\widehat{\tau}_{jj^{\prime}}\right]^{2},$
(3.4)

and $\widehat{\tau}_{jj^{\prime}}$ is defined in (2.3). Furthermore, with this
threshold, the sure screening property of Theorem 1 still holds.

The estimator (3.4) is a jackknife estimator for the asymptotic variance of a
U-statistic (Arvesen,, 1969; Sen,, 1977; Callaert and Veraverbeke,, 1981;
Fligner and Rust,, 1983; Lee,, 1985).

###  3.3 A Second Look at Assumptions 1 and 3

Assumptions 1 and 3 involve placing conditions on the elements of $\Sigma$
corresponding to non-zero and zero elements of $\Sigma^{-1}$, respectively.
These conditions are somewhat hard to interpret, since in general there is no
simple relationship between the $(j,j^{\prime})$th elements of a matrix $A$
and its inverse $A^{-1}$. We now present a result from Luo et al., (2015) that
allows us to re-formulate these assumptions as conditions on the elements of
$\Sigma^{-1}$. We let $\beta=\Lambda_{\max}(\Sigma)/\Lambda_{\min}(\Sigma)$
and
$\nu={2}/\left\\{\Lambda_{\max}(\Sigma)^{-1}+\Lambda_{\min}(\Sigma)^{-1}\right\\}$.
Here, $\Lambda_{\max}(\cdot)$ and $\Lambda_{\min}(\cdot)$ indicate the largest
and smallest eigenvalues of a matrix, respectively.

######  Proposition 1.

(Luo et al.,, 2015) Suppose
$1<\beta\leq\left\\{n^{(1-\xi)/2}+\Lambda_{\max}(\Sigma)^{-1/2}\right\\}/\left\\{n^{(1-\xi)/2}-\Lambda_{\max}(\Sigma)^{-1/2}\right\\}$.
Then Assumption 3 holds. Suppose also that
$n\geq\left(2/C_{1}\right)^{1/(1-\xi-\kappa)}$. If
$\min_{(j,j^{\prime})\in\mathcal{E}}\nu^{2}\left|(\Sigma^{-1})_{jj^{\prime}}\right|\geq
2C_{1}n^{-\kappa}$, then Assumption 1 holds. Furthermore, if Assumption 1
holds, then
$\min_{(j,j^{\prime})\in\mathcal{E}}\nu^{2}\left|(\Sigma^{-1})_{jj^{\prime}}\right|\geq
C_{1}n^{-\kappa}/2$.

Therefore, if $\Sigma$ is very well-conditioned and the non-zero elements of
$\Sigma^{-1}$ are sufficiently large, then Proposition 1 implies that Theorems
1 and 3 hold.

##  4 Simulation Studies

The simulation studies in this section are largely based on those in Luo et
al., (2015).

###  4.1 Data Generation

Let $p$ be the number of features, and $n$ the number of observations.
Motivated by the simulation study of Luo et al., (2015), we considered four
ways of generating the edge set $\mathcal{E}$.

  * _Simulation A:_

For all $j<j^{\prime}$, we set $(j,j^{\prime})\in\mathcal{E}$ with probability
$0.01$. We then generated a $p\times p$ matrix $A$, where

$A_{jj^{\prime}}=A_{j^{\prime}j}=\begin{cases}1&\text{for $j=j^{\prime}$}\\\
\mathrm{Unif}[-0.3,0.7]&\text{for $(j,j^{\prime})\in\mathcal{E}$ }\\\
0&\text{otherwise}\end{cases}.$ (4.1)

Finally, we created a positive definite matrix ${\Sigma}^{-1}$,

${\Sigma}^{-1}={A}+(0.1-\Lambda_{\min}({A})){I},$ (4.2)

where $\Lambda_{\min}(A)$ is the smallest eigenvalue of $A$, and $I$ denotes
the $p\times p$ identity matrix.

  * _Simulation B:_

We partitioned the $p$ features into $10$ equally-sized and non-overlapping
sets, $C_{l}=\\{(l-1)p/10+1,\ldots,lp/10\\}$ for $l=1,\ldots,10$. For all
$j\in C_{l},j^{\prime}\in C_{l},j<j^{\prime}$, we set
$(j,j^{\prime})\in\mathcal{E}$. We then generated $A$ and $\Sigma^{-1}$
according to (4.1) and (4.2).

  * _Simulation C:_

For all $j\leq j^{\prime}$, we set $\rho_{jj^{\prime}}=0.3^{|j-j^{\prime}|}$.

  * _Simulation D:_

We partitioned the features into $p/10$ equally-sized and non-overlapping
sets, $C_{l}=\\{10(l-1)+1,\ldots,10l\\}$ for $l=1,\ldots,p/10$. Then for all
$j\in C_{l},j^{\prime}\in C_{l}$, we set
$(\Sigma^{-1})_{jj^{\prime}}=0.9^{|j-j^{\prime}|}$. All other elements of
$\Sigma^{-1}$ were set to zero.

$\Sigma$ was rescaled to have diagonal elements equal to 1. We then generated
observations from a $N(0,{\Sigma})$ distribution, and observations from a
multivariate $t$-distribution with $\theta$ degrees of freedom, mean zero, and
correlation $\Sigma$. After that, we applied four monotonic functions,
$\exp(x)$, $x^{3}$, $x^{5}$, $(x-1)^{3}$, to these observations with equal
probability; this process gave us nonparanormal-distributed observations and
transelliptical $t$-distributed observations.

###  4.2 Control of False Positive Rate

Theorem 3 states that under certain conditions, performing transelliptical
GRASS with
$\gamma_{n,jj^{\prime}}=\frac{\pi}{2}\widehat{\omega}_{jj^{\prime}}\Phi^{-1}(1-\frac{f}{p(p-1)})/\sqrt{n}$,
where $\widehat{\omega}_{jj^{\prime}}$ is of the form (3.4), leads to control
of the asymptotic expected false positive rate (FPR) at level $q\equiv
f/|\mathcal{E}^{c}|$. In Tables 1 and 2, we explore the control of the FPR in
finite samples, for nonparanormal and transelliptical $t$-distributed data.
The FPR (defined as FP/(FP+TN)) and false negative rate (FNR; defined as
FN/(TP+FN)) are reported for various values of the level of desired FPR
control, $q$. The size of the estimated edge set
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ is also reported. Here $n=100$ and
$p=1000$, and results are averaged over 250 simulated data sets.

Assumption 3 is the key for controlling the asymptotic expected false positive
rate. In Simulation B, both $\Sigma$ and ${\Sigma}^{-1}$ are block diagonal
with ten completely dense blocks. So Assumption 3 holds in Simulation B. In
Simulation D, all of the zero elements in the precision matrix ${\Sigma}^{-1}$
also correspond to zero elements in $\Sigma$, so that Assumption 3 holds
exactly. As expected, the FPR is controlled successfully in Simulations B and
D.

In contrast, in Simulations A and C, not all of the zero elements in the
precision matrix ${\Sigma}^{-1}$ correspond to zero elements in the
correlation matrix $\Sigma$. But Table 1 and Table 2 reveal that the FPR is
still controlled well in these settings. This is because Assumption 3 only
requires the elements in $\mathcal{E}^{c}$ to correspond to small, though not
necessarily zero, elements of $\Sigma$. This assumption holds for most of the
elements in $\mathcal{E}^{c}$ and $\mathcal{E}$ for Simulations A and C.
Therefore, the FPR is also well-controlled in Simulations A and C.

Table 1: False positive rate control for nonparanormal data using
$\gamma_{n,jj^{\prime}}=\frac{\pi}{2}\widehat{\omega}_{jj^{\prime}}\Phi^{-1}(1-\frac{f}{p(p-1)})/\sqrt{n}$,
where $\widehat{\omega}_{jj^{\prime}}$ is of the form (3.4) in Theorem 3.

Simulation A Simulation B Simulation C Simulation D $q$
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR 1e-04 542.46 3e-04 0.925 1611.6
2e-04 0.969 290.22 2e-04 0.819 920.38 2e-04 0.819 0.001 1579.76 0.0019 0.868
3472.04 0.0014 0.943 1071.96 0.0014 0.648 1580.36 0.0014 0.806 0.01 7426.36
0.013 0.762 10950.86 0.011 0.879 6125.34 0.011 0.382 6364.42 0.011 0.793 0.1
53548.1 0.104 0.551 59265.12 0.098 0.695 49926.3 0.098 0.111 49884.46 0.098
0.722 0.2 102830.74 0.202 0.446 108698.82 0.195 0.580 98455.7 0.196 0.055
98333.9 0.195 0.644 0.3 152029.24 0.301 0.368 157545.2 0.294 0.488 147546.26
0.294 0.031 147393.94 0.294 0.566 0.5 250794.72 0.500 0.244 254988.28 0.493
0.332 246989.82 0.493 0.012 246737.66 0.493 0.405

Table 2: False positive rate control for transelliptical $t$-distributed data
with 5 degrees of freedom using
$\gamma_{n,jj^{\prime}}=\frac{\pi}{2}\widehat{\omega}_{jj^{\prime}}\Phi^{-1}(1-\frac{f}{p(p-1)})/\sqrt{n}$,
where $\widehat{\omega}_{jj^{\prime}}$ is of the form (3.4) in Theorem 3.

Simulation A Simulation B Simulation C Simulation D $q$
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR
$|\widehat{\mathcal{E}}_{\gamma_{n}}|$ FPR FNR 1e-04 448.836 3e-04 0.941
1325.172 2e-04 0.975 240.52 2e-04 0.87 859.488 2e-04 0.833 0.001 1447.58
0.0018 0.889 3039.336 0.0015 0.952 1012.12 0.0015 0.721 1564.668 0.0015 0.813
0.01 7261.952 0.013 0.788 10340.736 0.011 0.892 6123.648 0.0112 0.466 6438.42
0.0111 0.794 0.1 53507.624 0.104 0.575 58758.12 0.099 0.713 50330.904 0.099
0.158 50297.052 0.099 0.721 0.2 102902.456 0.203 0.468 108367.908 0.197 0.598
99069.432 0.197 0.085 98909.556 0.197 0.642 0.3 152190.388 0.302 0.387
157358.224 0.295 0.504 148208.564 0.295 0.053 148021.74 0.295 0.563 0.5
250987.548 0.50 0.258 254984.984 0.494 0.344 247579.316 0.495 0.023 247400.096
0.494 0.404

###  4.3 Comparison to Existing Approaches

We first compare the performances of the graphical lasso (Friedman et al.,,
2008), neighborhood selection (Meinshausen and Bühlmann,, 2006), and
transelliptical GRASS on nonparanormal data generated from Simulations A–D,
with $n=50$ and $p=750$. Let $X$ denote the $n\times p$ simulated data matrix.
We let GL($\widehat{\Sigma}$) and NS($\widehat{\Sigma}$) denote the results of
the graphical lasso and the neighborhood selection applied to the estimated
correlation matrix $\widehat{\Sigma}=X^{T}X/n$; and we let
GL($\widehat{S}^{\tau}$) and NS($\widehat{S}^{\tau}$) denote the results of
the graphical lasso and neighborhood selection applied to
$\widehat{S}^{\tau}$, the Kendall’s tau estimator, defined in (2.4). Results
are shown in Figure 1.

Transelliptical GRASS outperforms GL($\widehat{S}^{\tau}$) and
NS($\widehat{S}^{\tau}$) in Simulation B. The sparsity patterns of $\Sigma$
and ${\Sigma}^{-1}$ are identical, so the assumptions underlying
transelliptical GRASS hold.

In Simulations A and C, most of the edges in $\mathcal{E}$ correspond to large
elements of $\Sigma$, and most of the non-edges in $\mathcal{E}$ correspond to
small elements of $\Sigma$. Consequently, transelliptical GRASS outperforms
GL($\widehat{S}^{\tau}$) and NS($\widehat{S}^{\tau}$).

Simulation D was designed to violate Assumption 1; most of the elements in the
edge set $\mathcal{E}$ correspond to zero values in the correlation matrix
$\Sigma$. Therefore transelliptical GRASS does not perform well in Simulation
D. But even in this undesirable setting, Figure 1 indicates that
GL($\widehat{S}^{\tau}$) and NS($\widehat{S}^{\tau}$) do not perform much
better than the transelliptical GRASS.

Not surprisingly, the original GRASS proposal (which involves thresholding
$\widehat{\Sigma}$), GL($\widehat{\Sigma}$), and NS($\widehat{\Sigma}$)
perform poorly, because they are designed for Gaussian data rather than
nonparanormal data.

Finally, we compare the performance of the aforementioned methods on Gaussian
data generated from Simulations A–D, with $n=50$ and $p=750$. Figure 2 shows
that the original GRASS proposal performs only slightly better than
transelliptical GRASS on Gaussian data. This suggests that when the underlying
distribution of the data is unknown, there is little cost (and potentially a
large gain) associated with performing transelliptical GRASS instead of GRASS.

Overall, Figures 1 and 2 suggest that in these four settings, transelliptical
GRASS performs competitively compared to some popular but computationally-
intensive procedures for estimating a precision matrix.

Figure 1:  For nonparanormally distributed data with $p=750$ and $n=50$, the
true positive and false positive rates are shown. Curves are obtained by
varying the tuning parameter for each method, and results are averaged over 20
simulated data sets.  Figure 2:  For Gaussian-distributed data with $p=750$
and $n=50$, the true positive and false positive rates are shown. Curves are
obtained by varying the tuning parameter for each method, and results are
averaged over 20 simulated data sets.

##  5 Application to Equities data

We examined the Yahoo! Finance stock price data, which is described in Liu et
al., 2012b , and available in the `huge` package in `R` on CRAN. The data
consists of 1258 daily closing prices for 452 stocks in the S$\&$P 500 index
between January 1, 2003 and January 1, 2008. The stocks are categorized into
10 Global Industry Classification Standard (GICS) sectors. Let $S_{ij}$ denote
the closing price of the $j$th stock on the $i$th day. We construct a
$1257\times 452$ data matrix $X$ such that
$X_{ij}=\log\left(S_{(i+1)j}/S_{ij}\right)$ for $i=1,\dots,1257$ and
$j=1,\dots,452$. We standardize each stock to have mean zero and standard
deviation one, as in Tan et al., (2015).

We applied transelliptical GRASS with $\gamma_{n,jj^{\prime}}$=0.5, 0.6, 0.7,
and GL($\widehat{S}^{\tau}$) with $\lambda$=0.5, 0.6, 0.7. Figure 3 indicates
that in all estimated graphs, stocks from the same GICS sector tend to be
highly-connected, indicating that both methods provide informative graph
estimates. Furthermore, plots $(c)$, $(d)$, $(e)$ and $(f)$ of Figure 3 also
show that the graph estimates from transelliptical grass and
GL($\widehat{S}^{\tau}$) are quite similar provided that
$\gamma_{n,jj^{\prime}}=\lambda$. In fact, the arguments in Witten et al.,
(2011) can be used to establish the fact that when
$\gamma_{n,jj^{\prime}}=\lambda$, the connected components of GRASS and
GL($\widehat{S}^{\tau}$) are identical.

Figure 3:  $(a)$: Transelliptical GRASS with $\gamma_{n,jj^{\prime}}=0.5$
(3336 edges). $(b)$: GL($\widehat{S}^{\tau}$) with $\lambda=0.5$ (2343 edges).
$(c)$: Transelliptical GRASS with $\gamma_{n,jj^{\prime}}=0.6$ (1036 edges).
$(d)$: GL($\widehat{S}^{\tau}$) with $\lambda=0.6$ (847 edges). $(e)$:
Transelliptical GRASS with $\gamma_{n,jj^{\prime}}=0.7$ (232 edges). $(f)$:
GL($\widehat{S}^{\tau}$) with $\lambda=0.7$ (222 edges). The nodes are colored
based on the GICS sector of the corresponding stock.

##  6 Discussion

In this paper, we have proposed transelliptical GRASS, a simple and efficient
procedure for recovering the structure of a high-dimensional transelliptical
graphical model. Transelliptical GRASS is a natural extension of the GRASS
proposal of Luo et al., (2015) to the non-normal setting. Transelliptical
GRASS shares the attractive theoretical and computational properties of the
original GRASS proposal. We have established that it performs almost as well
as methods that assume Gaussianity when the data are Gaussian, and much better
than methods that assume Gaussianity in the case of non-Gaussian data.
Therefore, in general, there is little cost to applying transelliptical GRASS
instead of the original GRASS proposal.

##  Appendix A Appendix

###  A.1 Proof of Theorem 1

######  Definition 4.

(Hoeffding, (1963), page 25) Let $W_{1},W_{2},\ldots,W_{n}$ be independent
random variables. For $m\leq n$, a one-sample U-statistic takes the form

$U=\frac{1}{n^{(m)}}\sum_{n,m}g\left(W_{i_{1}},\ldots,W_{i_{m}}\right),$ (A.1)

where $n^{(m)}=n\left(n-1\right)\ldots\left(n-m+1\right)$, and the sum
$\sum_{n,m}$ is taken over all $m$-tuples $i_{1},\ldots,i_{m}$ of distinct
positive integers not exceeding $n$.

######  Lemma 2.

(Hoeffding, (1963), page 25) If the function of g in (A.1) is bounded as
$a\leq g\left(W_{i_{1}},\ldots,W_{i_{m}}\right)\leq b$, then for any $t>0$ and
$m\leq n$, we have

$\Pr\left\\{\left|U-{\mathbb{E}}\left(U\right)\right|>t\right\\}\leq
2\exp\left\\{\frac{-2\lfloor n/m\rfloor t^{2}}{(b-a)^{2}}\right\\}.$

Recall from (2.3) that

$\widehat{\tau}_{jj^{\prime}}=\frac{2}{n(n-1)}\sum_{1\leq i\leq i^{\prime}\leq
n}\mathrm{sign}\left(\left(X_{ij}-X_{i^{\prime}j}\right)\left(X_{ij^{\prime}}-X_{i^{\prime}j^{\prime}}\right)\right).$

Let $W_{i}=(X_{ij},X_{ij^{\prime}})$, $i=1,\ldots,n$. Then the sample
estimator $\widehat{\tau}_{jj^{\prime}}$ is of the form (A.1) with $m=2$ and
$g\left(W_{i},W_{i^{\prime}}\right)=\mathrm{sign}\left(\left(X_{ij}-X_{i^{\prime}j}\right)\left(X_{ij^{\prime}}-X_{i^{\prime}j^{\prime}}\right)\right)$.
Therefore, we can apply Lemma 2 with $m=2$ and $-1\leq
g\left(W_{i},W_{i^{\prime}}\right)\leq 1$, which yields

$\Pr\left(\left|\widehat{\tau}_{jj^{\prime}}-\tau_{jj^{\prime}}\right|>\frac{2}{3\pi}C_{1}n^{-\kappa}\right)\leq
2\exp\left(-\frac{2\lfloor n/2\rfloor}{9\pi^{2}}C_{1}^{2}n^{-2\kappa}\right).$

Next, we notice that

$\displaystyle\Pr\left(\left|\widehat{S}^{\tau}_{jj^{\prime}}-\rho_{jj^{\prime}}\right|>\frac{1}{3}C_{1}n^{-\kappa}\right)$
$\displaystyle=$
$\displaystyle\Pr\left(\left|\sin\left(\frac{\pi}{2}\widehat{\tau}_{jj^{\prime}}\right)-\sin\left(\frac{\pi}{2}\tau_{jj^{\prime}}\right)\right|>\frac{1}{3}C_{1}n^{-\kappa}\right)$
(A.2) $\displaystyle\leq$
$\displaystyle\Pr\left(\left|\widehat{\tau}_{jj^{\prime}}-\tau_{jj^{\prime}}\right|>\frac{2}{3\pi}C_{1}n^{-\kappa}\right)$
$\displaystyle\leq$ $\displaystyle 2\exp\left(-\frac{2\lfloor
n/2\rfloor}{9\pi^{2}}C_{1}^{2}n^{-2\kappa}\right).$

The first equality results from directly applying Lemma 1 and our definition
of $\widehat{S}^{\tau}_{jj^{\prime}}$. The first inequality results from
applying the mean value theorem. It follows that

$\displaystyle\Pr\left(\mathcal{E}\nsubseteq\widehat{\mathcal{E}}_{\gamma_{n}}\right)$
$\displaystyle=$
$\displaystyle\Pr\left\\{\bigcup_{(j,j^{\prime})\in\mathcal{E}}\left(\left|\widehat{S}^{\tau}_{jj^{\prime}}\right|<\gamma_{n}\right)\right\\}$
$\displaystyle\leq$
$\displaystyle\sum_{(j,j^{\prime})\in\mathcal{E}}\Pr\left(\left|\widehat{S}^{\tau}_{jj^{\prime}}\right|<\frac{2}{3}C_{1}n^{-\kappa}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{(j,j^{\prime})\in\mathcal{E}}\Pr\left(\left|\widehat{S}^{\tau}_{jj^{\prime}}-\rho_{jj^{\prime}}\right|\geq\frac{1}{3}C_{1}n^{-\kappa}\right)$
$\displaystyle\leq$ $\displaystyle 2p^{2}\exp\left(-\frac{2\lfloor
n/2\rfloor}{9\pi^{2}}C_{1}^{2}n^{-2\kappa}\right).$

Here, the second inequality follows from Assumption 1, and the third
inequality from the fact that $\mid\mathcal{E}\mid\leq p^{2}$ along with
(A.2).

Therefore, we have shown that

$\Pr(\mathcal{E}\subseteq\widehat{\mathcal{E}}_{\gamma_{n}})\geq
1-2p^{2}\cdot\exp\left(-\frac{2\lfloor
n/2\rfloor}{9\pi^{2}}C_{1}^{2}n^{-2\kappa}\right).$

A similar argument can be used to establish that

$\Pr\left(\mathcal{E}_{j}\subseteq\widehat{\mathcal{E}}_{j,\gamma_{n}}\right)\geq
1-2p^{2}\cdot\exp\left(-\frac{2\lfloor
n/2\rfloor}{9\pi^{2}}C_{1}^{2}n^{-2\kappa}\right).$

Conversely, if
$\min_{(j,j^{\prime})\in\mathcal{E}}|\rho_{jj^{\prime}}|<C_{1}n^{-\kappa}/3$,
then there exists $(j^{\star},{j^{\prime}}^{\star})\in\mathcal{E}$ such that
$|\rho_{j^{\star}{j^{\prime}}^{\star}}|<C_{1}n^{-\kappa}/3$. This together
with (A.2) implies that

$\displaystyle\Pr(\mathcal{E}\subseteq\widehat{\mathcal{E}}_{\gamma_{n}})\leq\Pr\left\\{(j^{\star},{j^{\prime}}^{\star})\in\widehat{\mathcal{E}}_{\gamma_{n}}\right\\}\leq\Pr\left(\left|\widehat{S}^{\tau}_{j^{\star}{j^{\prime}}^{\star}}-\rho_{j^{\star}{j^{\prime}}^{\star}}\right|\geq
C_{1}n^{-\kappa}/3\right)\leq 2\exp\left(-\frac{2\lfloor
n/2\rfloor}{9\pi^{2}}C_{1}^{2}n^{-2\kappa}\right),$

so that the result holds.

###  A.2 Proof of Corollary 1

###### Proof.

It suffices to show that transelliptical GRASS will not result in edges
between $x_{j}^{(s)}$ and $x_{j^{\prime}}^{(t)}$ for all $s\not=t$ with high
probability.

This is the case when the event
$\left(\left|\widehat{S}^{\tau}_{jj^{\prime}}-\rho_{jj^{\prime}}\right|\leq\frac{1}{3}C_{1}n^{-\kappa}\right)$
holds for all $j\neq j^{\prime}$. As was shown in the proof of Theorem 1,
$\Pr\left\\{\bigcap_{j\neq
j^{\prime}}\left|\widehat{S}^{\tau}_{jj^{\prime}}-\rho_{jj^{\prime}}\right|\leq\frac{1}{3}C_{1}n^{-\kappa}\right\\}\geq
1-C_{4}\mathrm{exp}\left(-C_{5}n^{1-2\kappa}\right)$. ∎

###  A.3 Proof of Theorem 2

Let

$L_{j}=\left\\{j^{\prime}:j^{\prime}\neq
j,\left|\rho_{jj^{\prime}}\right|\geq\frac{1}{3}C_{1}n^{-\kappa}\right\\}$

and

$\Gamma_{j}=\bigcap_{j^{\prime}:j^{\prime}\neq
j}\left\\{\left|\widehat{S}^{\tau}_{jj^{\prime}}-\rho_{jj^{\prime}}\right|\leq\frac{1}{3}C_{1}n^{-\kappa}\right\\}.$

By definition,
$\widehat{\mathcal{E}}_{j,\gamma_{n}}=\left\\{j^{\prime}:j^{\prime}\neq
j,\left|\widehat{S}^{\tau}_{jj^{\prime}}\right|>\frac{2}{3}C_{1}n^{-\kappa}\right\\}$.
On the set $\Gamma_{j}$, if $j^{\prime}$ belongs to
$\widehat{\mathcal{E}}_{j,\gamma_{n}}$, it has to belong to $L_{j}$. Thus, we
conclude that $\Pr\left(\widehat{\mathcal{E}}_{j,\gamma_{n}}\subseteq
L_{j}\right)\geq\Pr\left(\Gamma_{j}\right)$. An argument similar to that in
the proof of Theorem 1 can be used to show that

$\Pr\left(\Gamma_{j}\right)\geq 1-C_{4}\exp\left(-C_{5}n^{1-2\kappa}\right).$

This implies that

$\Pr\left(\widehat{\mathcal{E}}_{j,\gamma_{n}}\subseteq L_{j}\right)\geq
1-C_{4}\exp\left(-C_{5}n^{1-2\kappa}\right).$ (A.3)

Define $D=\sum_{j^{\prime}\in L_{j}}\rho_{jj^{\prime}}^{2}$. Then, by the
definition of $L_{j}$, it follows that

$D\geq\frac{1}{9}C_{1}^{2}\left|L_{j}\right|n^{-2\kappa}.$ (A.4)

Furthermore,

$D\leq\sum_{j^{\prime}=1}^{p}\rho_{jj^{\prime}}^{2}=\left\|\Sigma
e_{j}\right\|_{2}^{2}\leq\Lambda_{\max}\left(\Sigma\right)e_{j}^{T}\Sigma
e_{j}=\Lambda_{\max}\left(\Sigma\right),$ (A.5)

where $e_{j}$ is the unit vector with a one in the $j$th element and zeros
elsewhere, and where the last equality results from the fact that the diagonal
elements of $\Sigma$ are equal to $1$.

Combining (A.4) and (A.5), we have that

$|L_{j}|\leq 9C_{1}^{-2}n^{2\kappa}\Lambda_{\max}\left(\Sigma\right).$

This, in conjunction with Assumption 2 and (A.3), completes the proof of
Theorem 2.

###  A.4 Proof of Theorem 3

First, we verify the sure screening property (Section A.4.1). We then
establish the control of the asymptotic expected false positive rate (Section
A.4.2).

####  A.4.1 Verification of Sure Screening Property

To show that the sure screening property holds, it is enough to show that

$\gamma_{n,jj^{\prime}}=\frac{\pi}{2}\widehat{\omega}_{jj^{\prime}}\Phi^{-1}\left(1-\frac{f}{p(p-1)}\right)/\sqrt{n}\leq\frac{2}{3}C_{1}n^{-\kappa}.$
(A.6)

In other words, we must show that

$\frac{f}{p(p-1)}\geq
1-\Phi\left(\frac{4}{3\pi\widehat{\omega}_{jj^{\prime}}}C_{1}n^{\frac{1}{2}-\kappa}\right).$
(A.7)

Recall that
$1-\Phi\left(x\right)\leq\frac{1}{\sqrt{2\pi}}x^{-1}\exp\left(-x^{2}/2\right)$,
which implies that

$1-\Phi\left(\frac{4}{3\pi\widehat{\omega}_{jj^{\prime}}}C_{1}n^{\frac{1}{2}-\kappa}\right)\leq
C_{8}n^{-\frac{1}{2}+\kappa}\exp\left(-C_{9}n^{1-2\kappa}\right).$ (A.8)

Since $\log\left(p\right)=C_{3}n^{\xi}$, we have that

$f/\left\\{p(p-1)\right\\}\geq C_{10}\exp\left(-C_{11}n^{\xi}\right).$ (A.9)

Combining (A.8) and (A.9), and using the fact that $\xi<1-2\kappa$, (A.7)
follows directly.

####  A.4.2 Control of the Asymptotic Expected False Positive Rate

Next, we show that the choice of $\gamma_{n,jj^{\prime}}$ given in the
statement of Theorem 3 leads to control of the asymptotic expected false
positive rate at $f/|\mathcal{E}^{c}|$. The following lemma is used here.

######  Lemma 3.

Consider two random variables $X_{j}$ and $X_{j^{\prime}}$, each with $n$
i.i.d observations. Then

$\frac{\sqrt{n}(\widehat{\tau}_{jj^{\prime}}-\tau_{jj^{\prime}})}{\widehat{\omega}_{jj^{\prime}}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1),$ (A.10)

where $\widehat{\tau}_{jj^{\prime}}$ is defined in (2.3) and
$\widehat{\omega}_{jj^{\prime}}^{2}$ is defined in (3.4).

Lemma 3 follows from Theorem 6 in Arvesen, (1969) in conjunction with
Slutsky’s Theorem. It follows from an application of the delta method that

$\frac{\sqrt{n}\left(\widehat{S}_{jj^{\prime}}^{\tau}-\rho_{jj^{\prime}}\right)}{\frac{\pi}{2}\widehat{\omega}_{jj^{\prime}}\sqrt{\left(1-\rho^{2}_{jj^{\prime}}\right)}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1).$ (A.11)

Therefore, for any $(j,j^{\prime})\notin\mathcal{E}$, we have

$\displaystyle\Pr\left(\left|\widehat{S}_{jj^{\prime}}^{\tau}\right|>\gamma_{n,jj^{\prime}}\right)$
$\displaystyle=\Pr\left(\frac{\sqrt{n}\left(\widehat{S}_{jj^{\prime}}^{\tau}-\rho_{jj^{\prime}}\right)}{\frac{\pi\widehat{\omega}_{jj^{\prime}}}{2}\sqrt{1-\rho^{2}_{jj^{\prime}}}}>\frac{\sqrt{n}\left(\gamma_{n,jj^{\prime}}-\rho_{jj^{\prime}}\right)}{\frac{\pi\widehat{\omega}_{jj^{\prime}}}{2}\sqrt{1-\rho^{2}_{jj^{\prime}}}}\right)$
$\displaystyle\ \ \
+\Pr\left(\frac{\sqrt{n}\left(\widehat{S}_{jj^{\prime}}^{\tau}-\rho_{jj^{\prime}}\right)}{\frac{\pi\widehat{\omega}_{jj^{\prime}}}{2}\sqrt{1-\rho^{2}_{jj^{\prime}}}}<-\frac{\sqrt{n}\left(\gamma_{n,jj^{\prime}}+\rho_{jj^{\prime}}\right)}{\frac{\pi\widehat{\omega}_{jj^{\prime}}}{2}\sqrt{1-\rho^{2}_{jj^{\prime}}}}\right)$
$\displaystyle\to
1-\Phi\left(\Phi^{-1}\left(1-\frac{f}{p(p-1)}\right)\right)+1-\Phi\left(\Phi^{-1}\left(1-\frac{f}{p(p-1)}\right)\right)$
$\displaystyle=\frac{2f}{p(p-1)},$

where the convergence results from combining (A.11), Assumption 3, and the
fact that the order of $\sqrt{n}\gamma_{n,jj^{\prime}}$ is much larger than
that of $\sqrt{n}\rho_{jj^{\prime}}$ because $\sqrt{n}\gamma_{n,jj^{\prime}}$
is of the same order as $n^{\frac{\xi}{2}}$ while
$\sqrt{n}\rho_{jj^{\prime}}=o(n^{\frac{\xi}{2}})$ by Assumption 3.

Consequently, the expected $FPR$ is controlled as desired,

$\displaystyle{\mathbb{E}}(FPR)$
$\displaystyle=\frac{1}{|\mathcal{E}^{c}|}\sum_{\left(j,j^{\prime}\right)\notin\mathcal{E}}\Pr\left(\left|\widehat{S}_{jj^{\prime}}^{\tau}\right|>\gamma_{n,jj^{\prime}}\right)$
$\displaystyle\to\frac{\sum_{\left(j,j^{\prime}\right)\notin\mathcal{E}}\left(\frac{2f}{p(p-1)}\right)}{\left|\mathcal{E}^{c}\right|}=2f/\left(p\left(p-1\right)\right)\leq
f/|\mathcal{E}^{c}|,$

where the last inequality results from the fact that
$|\mathcal{E}^{c}|\leq\frac{p(p-1)}{2}$.

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