Measuring and testing interdependence among random vectors based on Spearman’s $\rho$ and Kendall’s $\tau$

Abstract

Inspired by the correlation matrix and based on the generalized Spearman’s $\rho$ and Kendall’s $\tau$ between random variables proposed in Lu et al. ( J Nonparametr Stat 30(4):860–883, 2018), $\rho$-matrix and $\tau$-matrix are suggested for multivariate data sets. The matrices are used to construct the $\rho$-measure and the $\tau$-measure among random vectors with statistical estimation and the asymptotic distributions under the null hypothesis of independence that produce the nonparametric tests of independence for multiple vectors. Simulation results demonstrate that the proposed tests are powerful under different grouping of the investigated random vector. An empirical application to detecting dependence of the closing price of a portfolio of stocks in NASDAQ also illustrates the applicability and effectiveness of our provided tests. Meanwhile, the corresponding measures are applied to characterize strength of interdependence of that portfolio of stocks during the recent two years.

Proof of Proposition 1

From (5), (8), for the couple $X_{k,r},X_{l,s}$, $k\ne l,r\in \{1,2,\ldots ,d_{k}\},s\in \{1,2,\ldots ,d_{l}\}$, the estimators of $\rho (X_{k,r},X_{l,s})$ and $\tau (X_{k,r},X_{l,s})$ by U-statistic are

$$\begin{aligned}&\hat{\rho }_{n}(X_{k,r},X_{l,s})\\&\quad =\frac{1}{C_{n}^{3}}\sum _{1\le i<j<t\le n}h_{\rho (X_{k,r},X_{l,s})}^{*}\left( (X_{k,r,i},X_{l,s,i}),(X_{k,r,j},X_{l,s,j}), (X_{k,r,t},X_{l,s,t})\right) , \end{aligned}$$

$$\begin{aligned} \hat{\tau }_{n}(X_{k,r},X_{l,s})=\frac{1}{C_{n}^{2}}\sum _{1\le i<j\le n}h_{\tau (X_{k,r},X_{l,s})}^{*}((X_{k,r,i},X_{l,s,i}),(X_{k,r,j},X_{l,s,j})), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&h_{\rho (X_{k,r},X_{l,s})}^{*}\left( (X_{k,r,i},X_{l,s,i}),(X_{k,r,j},X_{l,s,j}), (X_{k,r,t},X_{l,s,t})\right) \\&\quad =2\big (I_{\{X_{k,r,j}\le X_{k,r,i},X_{l,s,t}\le X_{l,s,i}\}}+I_{\{X_{k,r,t}\le X_{k,r,j},X_{l,s,i}\le X_{l,s,j}\}}+I_{\{X_{,k,r,i}\le X_{k,r,t},X_{l,s,j}\le X_{l,s,t}\}}\\&\qquad +I_{\{X_{k,r,t}\le X_{k,r,i},X_{l,s,j}\le X_{l,s,i}\}}+I_{\{X_{k,r,j}\le X_{k,r,t},X_{l,s,i}\le X_{l,s,t}\}}\\&\qquad +I_{\{X_{k,r,i}\le X_{k,r,j},X_{l,s,t}\le X_{l,s,j}\}}\big )-3, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned}&h_{\tau (X_{k,r},X_{l,s})}^{*}((X_{k,r,i},X_{l,s,i}),(X_{k,r,j},X_{l,s,j}))\\ =&\text {sign}((X_{k,r,i}-X_{k,r,j})(X_{l,s,i}-X_{l,s,j}))\\ =&2\left( I_{\{X_{k,r,i}>X_{k,r,j},X_{l,s,i}>X_{l,s,j}\}}+ I_{\{X_{k,r,i}<X_{k,r,j},X_{l,s,i}<X_{l,s,j}\}}\right) -1, \end{aligned} \end{aligned}$$

are the symmetric kernels of $\rho (X_{k,r},X_{l,s})$ and $\tau (X_{k,r},X_{l,s})$. Under the null hypothesis that $\varvec{X}^{(k)}$ and $\varvec{X}^{(l)},~k\ne l$ are independent,

$$\begin{aligned}&h_{\rho (X_{k,r},X_{l,s}),1}^{*}(x_{k,r,1},x_{l,s,1})\\\triangleq & {} E\left( h_{\rho (X_{k,r},X_{l,s})}^{*}\left( (X_{k,r,1},X_{l,s,1}),(X_{k,r,2},X_{l,s,2}), (X_{k,r,3},X_{l,s,3})\right) |(X_{k,r,1},X_{l,s,1})\right. \\&\left. =(x_{k,r,1},x_{l,s,1})\right) \\= & {} 4F_{k,r}(x_{k,r,1})F_{l,s}(x_{l,s,1})-2F_{k,r}(x_{k,r,1})-2F_{l,s}(x_{l,s,1})+1\\= & {} h_{\tau (X_{k,r},X_{l,s}),1}^{*}(x_{k,r,1},x_{l,s,1})\nonumber \\\triangleq & {} E\left( h_{\tau (X_{k,r},X_{l,s})}^{*}((X_{k,r,1},X_{l,s,1}),(X_{k,r,2},X_{l,s,2}))|(X_{k,r,1},X_{l,s,1}){=}(x_{k,r,1},x_{l,s,1})\right) ,\nonumber \\ \end{aligned}$$

then

$$\begin{aligned} E\left( h_{\xi (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,1},X_{l,s,1})\right) =0,~Var\left( h_{\xi (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,1},X_{l,s,1})\right) =\frac{1}{9}, \end{aligned}$$

(13)

where $\xi$ is $\rho$ or $\tau$. By Hoeffding (1948)’s decomposition theorem, we find

$$\begin{aligned} \sqrt{n}\hat{\rho }_{n}(X_{k,r},X_{l,s})\overset{\mathcal {L}}{\rightarrow }N(0,1),~ \sqrt{n}\hat{\tau }_{n}(X_{k,r},X_{l,s})\overset{\mathcal {L}}{\rightarrow }N(0,\frac{4}{9}),~~~n\rightarrow \infty . \end{aligned}$$

Let $W_{\rho (X_{k,r},X_{l,s}),n}=\frac{3}{\sqrt{n}}\sum _{i=1}^{n} h_{\rho (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,i},X_{l,s,i})$. Then

$$\begin{aligned}&E(\sqrt{n}\hat{\rho }_{n}(X_{k,r},X_{l,s})-W_{\rho (X_{k,r},X_{l,s}),n})^{2}\nonumber \\&=Var(\sqrt{n}\hat{\rho }_{n}(X_{k,r},X_{l,s}))+Var(W_{\rho (X_{k,r},X_{l,s}),n})\nonumber \\&\quad \,-6nE\left( \hat{\rho }_{n}(X_{k,r},X_{l,s})h_{\rho (X_{k,r},X_{l,s}),1}^{*}(\varvec{Z}_{k,r;l,s;1})\right) . \end{aligned}$$

(14)

Consider $E\left( h_{\rho (X_{k,r},X_{l,s})}^{*}\left( (X_{k,r,i},X_{l,s,i}),(X_{k,r,j},X_{l,s,j}), (X_{k,r,t},X_{l,s,t})\right) \right. \left. h_{\rho (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,i},X_{l,s,i})\right)$, where $1\le i<j<t\le n$. If $i=1$, from (13), for $\forall ~2\le j<t\le n$, we have

$$\begin{aligned}&E\left( E\left. \left( h_{\rho (X_{k,r},X_{l,s})}^{*}\left( (X_{k,r,1},X_{l,s,1}),(X_{k,r,j},X_{l,s,j}), (X_{k,r,t},X_{l,s,t})\right) \right. \right. \right. \nonumber \\&\quad \left. \left. \left. h_{\rho (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,1},X_{l,s,1})\Big | (X_{k,r,1},X_{l,s,1})\right) \right. \right) \nonumber \\&\quad =E\left( h_{\rho (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,1},X_{l,s,1})\right) ^{2}=\frac{1}{9}; \end{aligned}$$

(15)

If $i>1$, then

$$\begin{aligned}&E\left( h_{\rho (X_{k,r},X_{l,s})}^{*}\left( (X_{k,r,i},X_{l,s,i}),(X_{k,r,j},X_{l,s,j}), (X_{k,r,t},X_{l,s,t})\right) \right. \nonumber \\&\quad \left. h_{\rho (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,1},X_{l,s,1})\right) =0. \end{aligned}$$

(16)

Thus,

$$\begin{aligned} E\left( \hat{\rho }_{n}(X_{k,r},X_{l,s})h_{\rho (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,1},X_{l,s,1})\right) =\frac{1}{C_{n}^{3}}\cdot C_{n-1}^{2}\cdot \frac{1}{9}=\frac{1}{3n}. \end{aligned}$$

(17)

Since $\displaystyle \lim _{n\rightarrow \infty }Var(\sqrt{n}\hat{\rho }_{n}(X_{k,r},X_{l,s}))=Var(W_{\rho (X_{k,r},X_{l,s}),c})=1$, for $\forall ~c\ge 1$, combining (14)–(17), it is easy to see

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty }E(\sqrt{n}\hat{\rho }_{n}(X_{k,r},X_{l,s})-W_{\rho (X_{k,r},X_{l,s}),n})^{2}=0. \end{aligned}$$

(18)

Let $W_{\tau (X_{k,r},X_{l,s}),n}=\frac{2}{\sqrt{n}}\sum _{i=1}^{n}h_{\tau (X_{k,r},X_{l,s}),1}^{*}(X_{k,r,i},X_{l,s,i})$. According to the above procedure, we have

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty }E(\sqrt{n}\hat{\tau }_{n}(X_{k,r},X_{l,s})-W_{\tau (X_{k,r},X_{l,s}),n})^{2}=0 \end{aligned}$$

(19)

Using (18) and (19), for any $X_{k,r}$ from $\varvec{X}^{(k)}$, $X_{l,s}$ from $\varvec{X}^{(l)}$, $X_{a,\alpha }$ from $\varvec{X}^{(a)}$, and $X_{b,\beta }$ from $\varvec{X}^{(b)}$, $k\ne l,~a\ne b$, we have

$$\begin{aligned}&\displaystyle \lim _{n\rightarrow \infty }\left( \text {Cov}\left( \sqrt{n}\hat{\xi }_{n}(X_{k,r},X_{l,s}),\sqrt{n}\hat{\xi }_{n}(X_{a,\alpha },X_{b,\beta })\right) \right. \nonumber \\&\quad \left. -\text {Cov}\left( W_{\xi (X_{k,r},X_{l,s}),n},W_{\xi (X_{a,\alpha },X_{b,\beta }),n}\right) \right) =0, \end{aligned}$$

(20)

where $\xi$ denotes $\rho$ or $\tau$. Denote two $\sum _{k=1}^{m-1}\sum _{l=k+1}^{m}d_{k}d_{l}$-column vectors as

$$\begin{aligned} \varvec{\Lambda }_{\xi (\varvec{X}^{(1)},\ldots ,\varvec{X}^{(m)}),n}= \begin{pmatrix} \varvec{\Lambda }_{\xi (\varvec{X}^{(1)},\varvec{X}^{(2)}),n}\\ \varvec{\Lambda }_{\xi (\varvec{X}^{(1)},\varvec{X}^{(3)}),n}\\ \vdots \\ \varvec{\Lambda }_{\xi (\varvec{X}^{(1)},\varvec{X}^{(m)}),n}\\ \varvec{\Lambda }_{\xi (\varvec{X}^{(2)},\varvec{X}^{(3)}),n}\\ \vdots \\ \varvec{\Lambda }_{\xi (\varvec{X}^{(2)},\varvec{X}^{(m)}),n}\\ \vdots \\ \varvec{\Lambda }_{\xi (\varvec{X}^{(m-1)},\varvec{X}^{(m)}),n}\\ \end{pmatrix},~ \varvec{W}_{\xi (\varvec{X}^{(1)},\varvec{X}^{(2)},\ldots ,\varvec{X}^{(m)}),n}= \begin{pmatrix} \varvec{W}_{\xi (\varvec{X}^{(1)},\varvec{X}^{(2)}),n}\\ \varvec{W}_{\xi (\varvec{X}^{(1)},\varvec{X}^{(3)}),n}\\ \vdots \\ \varvec{W}_{\xi (\varvec{X}^{(1)},\varvec{X}^{(m)}),n}\\ \varvec{W}_{\xi (\varvec{X}^{(2)},\varvec{X}^{(3)}),n}\\ \vdots \\ \varvec{W}_{\xi (\varvec{X}^{(2)},\varvec{X}^{(m)}),n}\\ \vdots \\ \varvec{W}_{\xi (\varvec{X}^{(m-1)},\varvec{X}^{(m)}),n}\\ \end{pmatrix}, \end{aligned}$$

where $\varvec{\Lambda }_{\xi (\varvec{X}^{(k)},\varvec{X}^{(l)}),n}= \begin{pmatrix} \hat{\xi }_{n}(X_{k,1},X_{l,1})\\ \hat{\xi }_{n}(X_{k,1},X_{l,2})\\ \vdots \\ \hat{\xi }_{n}(X_{k,1},X_{l,d_{l}})\\ \hat{\xi }_{n}(X_{k,2},X_{l,1})\\ \vdots \\ \hat{\xi }_{n}(X_{k,2},X_{l,d_{l}})\\ \vdots \\ \hat{\xi }_{n}(X_{k,d_{k}},X_{l,d_{l}}) \end{pmatrix},~ \varvec{W}_{\xi (\varvec{X}^{(k)},\varvec{X}^{(l)}),n}= \begin{pmatrix} W_{\xi (X_{k,1},X_{l,1}),n}\\ W_{\xi (X_{k,1},X_{l,2}),n}\\ \vdots \\ W_{\xi (X_{k,1},X_{l,d_{l}}),n}\\ W_{\xi (X_{k,2},X_{l,1}),n}\\ \vdots \\ W_{\xi (X_{k,2},X_{l,d_{l}}),n}\\ \vdots \\ W_{\xi (X_{k,d_{k}},X_{l,d_{l}}),n} \end{pmatrix}$ are $d_{k}d_{l}$-vectors, $1\le k<l\le m$ and $\xi$ represents $\rho$ or $\tau$. From (18) and (19), it is proved that $\sqrt{n}\varvec{\Lambda }_{\xi (\varvec{X}^{(1)},\ldots ,\varvec{X}^{(m)}),n}$ and $\varvec{W}_{\xi (\varvec{X}^{(k)},\varvec{X}^{(l)}),n}$ have the same limit distribution. Combining (13) and multidimensional CLT, as $n\rightarrow \infty$, we have

$$\begin{aligned} \begin{aligned}&\varvec{W}_{\rho (\varvec{X}^{(1)},\varvec{X}^{(2)},\ldots ,\varvec{X}^{(m)}),n} \overset{\mathcal {L}}{\rightarrow }N_{\sum _{1\le k<l\le m}d_{k}d_{l}}(\varvec{0},\varvec{\Sigma }),\\&\varvec{W}_{\tau (\varvec{X}^{(1)},\varvec{X}^{(2)},\ldots ,\varvec{X}^{(m)}),n} \overset{\mathcal {L}}{\rightarrow }N_{\sum _{1\le k<l\le m}d_{k}d_{l}}(\varvec{0},\frac{4}{9}\varvec{\Sigma }), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \varvec{\Sigma }&=\begin{pmatrix} \varvec{\Sigma }_{1} &{} &{} &{} \\ &{} \varvec{\Sigma }_{2} &{} &{} \text {{0}} \\ &{} &{}\ddots &{} \\ &{} \text {{0}} &{} &{} \varvec{\Sigma }_{\frac{m(m-1)}{2}} \end{pmatrix},~~ \varvec{\Sigma }_{\frac{(2m-k)(k-1)}{2}+l-k}\nonumber \\&=\text {Cov}(\varvec{W}_{\rho (\varvec{X}^{(k)},\varvec{X}^{(l)})},\varvec{W}_{\rho (\varvec{X}^{(k)},\varvec{X}^{(l)})}),1\le k<l\le m \end{aligned}$$

(21)

is a symmetric and positive definite matrix of dimension $\left( \sum _{1\le k<l\le m}d_{k}d_{l}\right) \times \left( \sum _{1\le k<l\le m}d_{k}d_{l}\right)$, and for any $1\le k<l\le m$, $1\le a<b\le m$, $r\in \{1,\ldots ,d_{k}\}$, $s\in \{1,\ldots ,d_{l}\}$, $\alpha \in \{1,\ldots ,d_{a}\}$ and $\beta \in \{1,\ldots ,d_{b}\}$, the $(\sum _{\mu =1}^{k-1}\sum _{\nu =\mu +1}^{m}d_{\mu }d_{\nu } +d_{k}\sum _{\nu =k+1}^{l-1}d_{\nu }+(r-1)d_{l}+s,\sum _{\mu =1}^{a-1} \sum _{\nu =\mu +1}^{m}d_{\mu }d_{\nu }+d_{a}\sum _{\nu =a+1}^{b-1}d_{\nu }+ (\alpha -1)d_{b}+\beta )$ entry of $\varvec{\Sigma }$ is

$$\begin{aligned}&36\text {Cov}(2U_{k,r,i}U_{l,s,i}-U_{k,r,i}-U_{l,s,i}, 2U_{a,\alpha ,i}U_{b,\beta ,i}-U_{a,\alpha ,i}-U_{b,\beta ,i})\\&=\rho (X_{k,r,i},X_{a,\alpha ,i})\rho (X_{l,s,i},X_{b,\beta ,i}), \end{aligned}$$

where $U_{k,r,i}=F_{k,r}(X_{k,r,i})$. Note that $\rho _{s}(X_{k,r,i},X_{a,\alpha ,i})\rho _{s}(X_{l,s,i},X_{b,\beta ,i})=0$ unless $k=a,l=b$. Especially, the entries of $\varvec{\Sigma }$ on the principal diagonal are 1. Hence, as $n\rightarrow \infty$,

$$\begin{aligned} \begin{aligned}&\sqrt{n}\varvec{\Lambda }_{\rho (\varvec{X}^{(1)},\varvec{X}^{(2)},\ldots ,\varvec{X}^{(m)}),n} \overset{\mathcal {L}}{\rightarrow }N_{\sum _{1\le k<l\le m}d_{k}d_{l}}(\varvec{0},\varvec{\Sigma }),\\&\sqrt{n}\varvec{\Lambda }_{\tau (\varvec{X}^{(1)},\varvec{X}^{(2)},\ldots ,\varvec{X}^{(m)}),n} \overset{\mathcal {L}}{\rightarrow }N_{\sum _{1\le k<l\le m}d_{k}d_{l}}(\varvec{0},\frac{4}{9}\varvec{\Sigma }). \end{aligned} \end{aligned}$$

(22)

Let $\varvec{I}_{\sum _{1\le k<l\le m}d_{k}d_{l}}=(1,1,\ldots ,1)^{\top }$ be a vector of dimension $\sum _{1\le k<l\le m}d_{k}d_{l}$. Rewrite

$$\begin{aligned} \sqrt{n}\hat{D}_{\xi _{n}}(\varvec{X}^{(1)},\ldots ,\varvec{X}^{(m)}) =\frac{\sqrt{n}}{\sum _{1\le k<l\le m}d_{k}d_{l}}\varvec{I}^{\top }_{\sum _{1\le k<l\le m}d_{k}d_{l}}\cdot \varvec{\Lambda }_{\xi (\varvec{X}^{(1)},\ldots ,\varvec{X}^{(m)})}. \end{aligned}$$

(23)

Since

$$\begin{aligned}&\varvec{I}^{\top }_{\sum _{1\le k<l\le m}d_{k}d_{l}}\cdot \varvec{\Sigma }\cdot \varvec{I}_{\sum _{1\le k<l\le m}d_{k}d_{l}}\nonumber \\= & {} \sum _{k,a=1}^{m-1}\sum _{l=k+1}^{m}\sum _{b=a+1}^{m} \varvec{I}_{d_{k}d_{l}}^{\top }\cdot \text {Cov}(\sqrt{n}\varvec{\Lambda }_{\rho (\varvec{X}^{(k)},\varvec{X}^{(l)}),n},\sqrt{n}\varvec{\Lambda }_{\rho (\varvec{X}^{(a)},\varvec{X}^{(b)}),n})\cdot \varvec{I}_{d_{a}d_{b}}\nonumber \\= & {} \sum _{k=1}^{m-1}\sum _{l=k+1}^{m} \varvec{I}_{d_{k}d_{l}}^{\top }\cdot \text {Cov}(\sqrt{n}\varvec{\Lambda }_{\rho (\varvec{X}^{(k)},\varvec{X}^{(l)}),n},\sqrt{n}\varvec{\Lambda }_{\rho (\varvec{X}^{(k)},\varvec{X}^{(l)}),n})\cdot \varvec{I}_{d_{k}d_{l}}\nonumber \\= & {} \sum _{k=1}^{m-1}\sum _{l=k+1}^{m}\sum _{r,\alpha =1}^{d_{k}}\sum _{s,\beta =1}^{d_{l}}\rho _{s}(X_{k,r},X_{k,\alpha })\rho _{s}(X_{l,s},X_{l,\beta })\\= & {} \sum _{k=1}^{m-1}\sum _{l=k+1}^{m}\sum _{r,\alpha =1}^{d_{k}}\sum _{s,\beta =1}^{d_{l}} \text {Cov}(F_{k,r}(X_{k,r}),F_{k,\alpha }(X_{k,\alpha }))\text {Cov}(F_{l,s}(X_{l,s}),F_{l,\beta }(X_{l,\beta }))\nonumber \\= & {} \sum _{k=1}^{m-1}\sum _{l=k+1}^{m} \text {Cov}(\sum _{r=1}^{d_{k}}F_{k,r}(X_{k,r}),\sum _{\alpha =1}^{d_{k}}F_{k,\alpha }(X_{k,\alpha })) \text {Cov}(\sum _{s=1}^{d_{l}}F_{l,s}(X_{l,s}),\sum _{\beta =1}^{d_{l}}F_{l,\beta }(X_{l,\beta }))\nonumber \\= & {} \sum _{k=1}^{m-1}\sum _{l=k+1}^{m} Var(\sum _{r=1}^{d_{k}}F_{k,r}(X_{k,r}))Var(\sum _{s=1}^{d_{l}}F_{l,s}(X_{l,s}))>0,\nonumber \end{aligned}$$

(24)

we note that the third equality holds since (21) in which $\varvec{\Sigma }$ has $\varvec{\Sigma }_{i},i=1,\ldots ,\frac{m(m-1)}{2}$ down the principal diagonal and zeros elsewhere, and the results follow from (22)–(24).