Tail Maximal Dependence in Bivariate Models: Estimation and Applications

Abstract

Assessing dependence within co-movements of financial instruments has been of much interest in risk management. Typically, indices of tail dependence are used to quantify the strength of such dependence, although many of them underestimate the strength. Hence, we advocate the use of indices of maximal tail dependence, and for this reason we also develop a statistical procedure for estimating the indices. We illustrate the procedure using simulated and real data sets.

Appendices

APPENDIX

A TECHNICALITIES

The proof of Theorem 3.1 is long, but it is necessary to present in order to see why and how the estimator works. We note at the outset that the joint cdf $F_{q}^{*}$ may not have the uniform on $[0,1]$ marginal distributions, and so bound (3.13) does not really mean that $F_{q}^{*}$ is PQD. Nevertheless, it is this bound that we need in the proof of Theorem 3.1. In the following appendices we shall test the validity of this bound using a number of statistical tests.

2.1 A.1. Proof of Theorem 3.1

We start the proof by expressing $\kappa^{*}(m,0,q)$ in terms of the survival function of the random variable

$$\xi_{q}:={2\log(C(U_{q},V_{q})/\Pi^{*}(q))\over\log(U_{q}V_{q}/q^{2})},$$

(A.1)

where $(U_{q},V_{q})$ follows the cdf $F_{q}$. That is, we have the equation

$$\kappa^{*}(m,0,q)=\int\limits_{0}^{\infty}\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x$$

(A.2)

because the cardinality of the set $G_{1,q}$ is $m$ and the pairs $(U_{i,q},V_{i,q})$, $i\in G_{1,q}$, are iid. With the notation

$$(\widetilde{U}_{q},\widetilde{V}_{q}):=\bigg{(}{U_{q}\over\varphi^{*}(q)},{V_{q}\varphi^{*}(q)\over q^{2}}\bigg{)}$$

we rewrite $\xi_{q}$ as

$$\xi_{q}={2\log(C(U_{q},V_{q})/\Pi^{*}(q))\over\log(U_{q}/\varphi^{*}(q))+\log(V_{q}\varphi^{*}(q)/q^{2})}={2\log F_{q}^{*}(\widetilde{U}_{q},\widetilde{V}_{q})\over\log\widetilde{U}_{q}+\log\widetilde{V}_{q}}\in[0,2],$$

(A.3)

where the inclusion into the interval $[0,2]$ is due to the assumed bound (3.13). Hence, bound (A.2) reduces to

$$\kappa^{*}(m,0,q)=\int\limits_{0}^{2}\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x.$$

(A.4)

Note A.1. The intuitive meaning of Eq. (A.3) is to scale the pair $(U_{q},V_{q})\in\mathcal{R}_{q}(\mathbf{0})$ into $(\widetilde{U}_{q},\widetilde{V}_{q})\in[0,1]\times[0,1]$. This allows us to shift the focus from the behaviour of random pairs with respect to $q\downarrow 0$ toward the behaviour of $F_{q}^{*}$ and the scaling parameters $\varphi^{*}(q)$ and $q^{2}/\varphi^{*}(q)$.

Since $(U_{q},V_{q})\in\mathcal{R}_{q}(\mathbf{0})$ and the functions $\varphi^{*}$ and $z\mapsto z^{2}/\varphi^{*}(z)$ are increasing, we have

$$(\widetilde{U}_{q}^{*},\widetilde{V}_{q}^{*}):=\bigg{(}{\varphi^{*}(\sqrt{U_{q}V_{q}})\over\varphi^{*}(q)},{(\sqrt{U_{q}V_{q}})^{2}/\varphi^{*}(\sqrt{U_{q}V_{q}})\over q^{2}/\varphi^{*}(q)}\bigg{)}\in[0,1]\times[0,1].$$

Hence,

$$F_{q}^{*}(\widetilde{U}_{q}^{*},\widetilde{V}_{q}^{*})={1\over\Pi^{*}(q)}C\big{(}\varphi^{*}(\sqrt{U_{q}V_{q}}),U_{q}V_{q}/\varphi^{*}(\sqrt{U_{q}V_{q}})\big{)}$$

$${}={1\over\Pi^{*}(q)}\sup_{x\in[U_{q}V_{q},1]}C\big{(}x,U_{q}V_{q}/x)\big{)}\geq{1\over\Pi^{*}(q)}C(U_{q},V_{q})=F_{q}^{*}(\widetilde{U}_{q},\widetilde{V}_{q}).$$

Since $\widetilde{U}_{q}^{*}\widetilde{V}_{q}^{*}=\widetilde{U}_{q}\widetilde{V}_{q}=U_{q}V_{q}/q^{2}\in(0,1)$, we arrive at the bound

$$\dfrac{2\log F_{q}^{*}(\widetilde{U}_{q}^{*},\widetilde{V}_{q}^{*})}{\log\widetilde{U}_{q}^{*}+\log\widetilde{V}_{q}^{*}}\leq\dfrac{2\log F_{q}^{*}(\widetilde{U}_{q},\widetilde{V}_{q})}{\log\widetilde{U}_{q}+\log\widetilde{V}_{q}}.$$

(A.5)

Note that

$${2\log F_{q}^{*}(\widetilde{U}_{q}^{*},\widetilde{V}_{q}^{*})\over\log\widetilde{U}_{q}^{*}+\log\widetilde{V}_{q}^{*}}={2\log\Big{(}C\big{(}\varphi^{*}(\sqrt{U_{q}V_{q}}),(\sqrt{U_{q}V_{q}})^{2}/\varphi^{*}(\sqrt{U_{q}V_{q}})\big{)}/\Pi^{*}(q)\Big{)}\over\log(U_{q}V_{q}/q^{2})}$$

$${}={2\log\big{(}\Pi^{*}(\sqrt{U_{q}V_{q}})/\Pi^{*}(q)\big{)}\over\log(U_{q}V_{q}/q^{2})}.$$

(A.6)

Due to Eq. (2.1), we have

$${\Pi^{*}(\sqrt{U_{q}V_{q}})\over\Pi^{*}(q)}={\ell^{*}(\sqrt{U_{q}V_{q}})\over\ell^{*}(q)}\bigg{(}{U_{q}V_{q}\over q^{2}}\bigg{)}^{\kappa^{*}/2}$$

and thus, continuing with Eq. (A.6) and taking into account bound (A.5), we obtain

$$\dfrac{2\log F_{q}^{*}(\widetilde{U}_{q},\widetilde{V}_{q})}{\log\widetilde{U}_{q}+\log\widetilde{V}_{q}}\geq{2\log(\ell^{*}(\sqrt{U_{q}V_{q}})/\ell^{*}(q))\over\log(U_{q}V_{q}/q^{2})}+\kappa^{*}.$$

(A.7)

With the notation

$$o_{q}:={2\log(\ell^{*}(\sqrt{U_{q}V_{q}})/\ell^{*}(q))\over\log(U_{q}V_{q}/q^{2})}$$

bound (A.7) takes the form

$$\xi_{q}\geq\max\{0,o_{q}+\kappa^{*}\}.$$

(A.8)

We next prove the statement

$$o_{q}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}0$$

(A.9)

when $q\downarrow 0$. With the notation $W_{q}=\sqrt{U_{q}V_{q}}/q$, statement (A.9) is equivalent to

$$o_{q}={\log(\ell^{*}(qW_{q})/\ell^{*}(q))\over\log W_{q}}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}0.$$

To prove it, we fix $\varepsilon>0$ and $\delta>0$, with the latter parameter used to partition the sample space into the following three events: $\{W_{q}<\delta\}$, $\{\delta\leq W_{q}\leq 1-\delta\}$ and $\{W_{q}>1-\delta\}$. We obtain the bound

$$\mathbb{P}\bigg{(}\bigg{|}{\log(\ell^{*}(qW_{q})/\ell^{*}(q))\over\log W_{q}}\bigg{|}>\varepsilon\bigg{)}$$

$${}\leq\mathbb{P}\bigg{(}\sup_{w\in[\delta,1-\delta]}\bigg{|}{\log(\ell^{*}(qw)/\ell^{*}(q))\over\log w}\bigg{|}>\varepsilon\bigg{)}+\mathbb{P}\left(W_{q}<\delta\right)+\mathbb{P}\left(W_{q}>1-\delta\right).$$

(A.10)

Since $\ell^{*}$ is slowly varying at $0$, we have $\ell^{*}(qw)/\ell^{*}(q)\to 1$ when $q\downarrow 0$ for every $w\in(0,1]$. The convergence is uniform in $w\in[\delta,1]$ for any fixed $\delta>0$, which implies [6, Lemma 1, p. 310] that $\sup_{w\in[\delta,1]}|\log(\ell^{*}(qw)/\ell^{*}(q))|$ converges to $0$ when $q\downarrow 0$. Hence, we conclude from bound (A.10) that, for any $\varepsilon>0$ and $\delta>0$,

$$\limsup_{q\to 0}\mathbb{P}\bigg{(}\bigg{|}{\log(\ell^{*}(qW_{q})/\ell^{*}(q))\over\log W_{q}}\bigg{|}>\varepsilon\bigg{)}$$

$${}\leq\limsup_{q\to 0}\mathbb{P}\left(W_{q}<\delta\right)+\limsup_{q\to 0}\mathbb{P}\left(W_{q}>1-\delta\right).$$

(A.11)

Note that the left-hand side of bound (A.11) does not depend on $\delta>0$. As to the first probability on the right-hand side of bound (A.11), we have

$$\mathbb{P}(W_{q}\leq\delta)=\mathbb{P}(U_{q}V_{q}\leq\delta\varphi^{*}(q)\delta q^{2}/\varphi^{*}(q))\leq\mathbb{P}(U_{q}\leq\delta\varphi^{*}(q))+\mathbb{P}(V_{q}\leq\delta q^{2}/\varphi^{*}(q))$$

$${}=\dfrac{1}{\Pi^{*}(q)}\left(C(\delta\varphi^{*}(q),q^{2}/\varphi^{*}(q))+C(\varphi^{*}(q),\delta q^{2}/\varphi^{*}(q))\right)\leq\dfrac{2\Pi^{*}(q\sqrt{\delta})}{\Pi^{*}(q)}\to 2\delta^{\kappa^{*}/2}$$

(A.12)

when $q\downarrow 0$, where we used equation (2.1) and property (1.2).

We tackle the second probability on the right-hand side of bound (A.11) in a different way, starting as follows:

$$\mathbb{P}(W_{q}>1-\delta)=\mathbb{P}(\widetilde{U}_{q}\widetilde{V}_{q}>(1-\delta)^{2})\leq\mathbb{P}(F^{*}_{q}(\widetilde{U}_{q},\widetilde{V}_{q})>(1-\delta)^{2}),$$

(A.13)

where the bound holds because $F^{*}_{q}(u,v)\geq uv$ for all $u,v\in[0,1]$. With the notation

$$K^{*}_{q}(t):=\mathbb{P}(F^{*}_{q}(\widetilde{U}_{q},\widetilde{V}_{q})\leq t)$$

we continue with bound (A.13) and have

$$\mathbb{P}(W_{q}>1-\delta)=1-K^{*}_{q}((1-\delta)^{2})\leq 1-(1-\delta)^{2}$$

(A.14)

for every $q\in(0,1]$, because

$$K^{*}_{q}(t)\geq\mathbb{P}(F^{*}_{q}(\widetilde{U}_{q},1)\leq t)=\mathbb{P}(G_{q}(\widetilde{U}_{q})\leq t)$$

$${}=\mathbb{P}(\widetilde{U}_{q}\leq G^{-1}_{q}(t))=G_{q}\circ G^{-1}_{q}(t)=t,$$

where $G_{q}:[0,1]\to[0,1]$ denotes the cdf of $\widetilde{U}_{q}$ given by

$$G_{q}(u)=\dfrac{C(\varphi^{*}(q)u,q^{2}/\varphi^{*}(q))}{C(\varphi^{*}(q),q^{2}/\varphi^{*}(q))}=\dfrac{C(\varphi^{*}(q)u,q^{2}/\varphi^{*}(q))}{\Pi^{*}(q)}.$$

Hence, in view of bounds (A.12) and (A.14), the entire right-hand side of bound (A.11) vanishes when $\delta\downarrow 0$. This concludes the proof of statement (A.9).

Fix now any $\varepsilon\in(0,\kappa^{*})$, which, by the way, has nothing to do with the earlier used $\varepsilon$. Equation (A.4) implies the bound

$$\kappa^{*}(m,0,q)\leq\kappa^{*}+\varepsilon+\int\limits_{\kappa^{*}+\varepsilon}^{2}\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x.$$

(A.15)

To estimate $\kappa^{*}(m,0,q)$ from below, we start with

$$\kappa^{*}(m,0,q)=\kappa^{*}-\varepsilon-\int\limits_{0}^{\kappa^{*}-\varepsilon}1-\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x+\int\limits_{\kappa^{*}+\varepsilon}^{2}\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x$$

$${}\geq\kappa^{*}-\varepsilon-(\kappa^{*}-\varepsilon)m\mathbb{P}(\xi_{q}\leq\kappa^{*}-\varepsilon)+\int\limits_{\kappa^{*}+\varepsilon}^{2}\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x.$$

(A.16)

Using bound (A.8), we obtain

$$\mathbb{P}(\xi_{q}\leq\kappa^{*}-\varepsilon)\leq\mathbb{P}\big{(}\max\{0,o_{q}+\kappa^{*}\}\leq\kappa^{*}-\varepsilon\big{)}$$

$${}=\mathbb{P}\big{(}o_{q}\leq-\varepsilon,o_{q}+\kappa^{*}>0\big{)}+\mathbb{P}\big{(}o_{q}+\kappa^{*}\leq 0\big{)}=2\mathbb{P}(|o_{q}|\geq\varepsilon).$$

(A.17)

The right-hand side can be made as small as desired by choosing a sufficiently small $q>0$. Hence, due to bounds (A.15)–(A.17), for any $m\geq 1$ we can choose sufficiently small $\varepsilon>0$ and $q>0$ such that $\kappa^{*}(m,0,q)$ is as close to $\kappa^{*}$ as desired, provided that the integral $\int_{\kappa^{*}}^{2}\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x$ can be made as small as desired by choosing a sufficiently large $m$.

To prove the latter statement, without loss of generality we assume $\kappa^{*}<2$, which prevents $F_{q}^{*}$ from being the independence copula. By the Lebesgue dominated convergence theorem, the integral $\int_{\kappa^{*}}^{2}\mathbb{P}(\xi_{q}>x)^{m}\textrm{d}x$ converges to $0$ when $m\to\infty$ provided that $\mathbb{P}(\xi_{q}>x)<1$ for all $x\in(\kappa^{*},2)$. Hence, we need to show that

$$\mathbb{P}(\xi_{q}\leq\kappa^{*}+h)>0\quad\text{for every}\quad h\in(0,2-\kappa^{*}).$$

(A.18)

Although the proof of this statement follows the ideas of Sun et al. [37, Theorem 3.2], substantial adjustments are required, which we give next.

We start with the bound

$$\mathbb{P}(\xi_{q}\leq\kappa^{*}+h)=\mathbb{P}\bigg{(}\dfrac{2\log F_{q}^{*}(\widetilde{U}_{q},\widetilde{V}_{q})}{\log\widetilde{U}_{q}+\log\widetilde{V}_{q}}\leq\dfrac{2\log w^{\kappa^{*}+h}_{0}}{\log w^{2}_{0}}\bigg{)}\geq\mathbb{P}\left((\widetilde{U}_{q},\widetilde{V}_{q})\in B_{q,h}\right),$$

(A.19)

where, for some $w_{0}\in(0,1)$,

$$B_{q,h}:=\big{\{}(u,v)\in[0,1]\times[0,1]:uv\leq w^{2}_{0},F_{q}^{*}(u,v)>w^{\kappa^{*}+h}_{0}\big{\}},$$

which is depicted in Fig. 6. With the notation $w_{0}=\sqrt{u_{q}v_{q}}$, we have

$$u_{q}=\varphi^{*}(qw_{0})/\varphi^{*}(q),\quad v_{q}=w^{2}_{0}\varphi^{*}(q)/\varphi^{*}(qw_{0}),$$

and

$$F_{q}^{*}(u_{q},v_{q})=\dfrac{\ell^{*}(qw_{0})}{\ell^{*}(q)}w^{\kappa^{*}}_{0}\approx w^{\kappa^{*}}_{0}>w^{\kappa^{*}+h}_{0}.$$

Fig. 6

The set $B_{q,h}$ and associated curves.

Hence, $F_{q}^{*}(u_{q},v_{q})>w^{\kappa^{*}+h}_{0}$ for a sufficiently small $q>0$ (depending on $w_{0}$ and $h$). Note that

$$F_{q}^{*}\bigg{(}\dfrac{\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0})}{\varphi^{*}(q)},\dfrac{w^{2+h/\kappa^{*}}_{0}\varphi^{*}(q)}{\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0})}\bigg{)}=\dfrac{\Pi^{*}(qw^{1+h/2\kappa^{*}}_{0})}{\Pi^{*}(q)}\approx w^{\kappa^{*}+h/2}_{0}.$$

We have

$$F_{q}^{*}\bigg{(}\dfrac{\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0})}{\varphi^{*}(q)},\dfrac{w^{2+h/\kappa^{*}}_{0}\varphi^{*}(q)}{\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0})}\bigg{)}>w^{\kappa^{*}+h}_{0}$$

for as sufficiently small $q>0$. With

$$u_{q,h}:=\dfrac{\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0})}{\varphi^{*}(q)}\quad\textrm{and}\quad v_{q,h}:=\dfrac{w^{2+h/\kappa^{*}}_{0}\varphi^{*}(q)}{\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0})},$$

we have

$$u_{q,h}v_{q,h}=w^{2+h/\kappa^{*}}_{0}<w^{2}_{0}.$$

Thus, $(u_{q,h},v_{q,h})\in B_{q,h}$. In particular, $(u_{q,0},v_{q,0})=(u_{q},v_{q})$. Since $\varphi^{*}$ and $\psi^{*}$ are strictly increasing functions, we have $u_{q,h}<u_{q}$ and $v_{q,h}<v_{q}$, and so the rectangle

$$E_{q,h}:=(u_{q,h},u_{q}]\times(v_{q,h},v_{q}],$$

which we depict in Fig. 7, is a non-empty subset of $B_{q,h}$. We have

$$\mathbb{P}\big{(}(\widetilde{U}_{q},\widetilde{V}_{q})\in E_{q,h}\big{)}=F_{q}^{*}(u_{q},v_{q})-F_{q}^{*}(u_{q},v_{q,h})-F_{q}^{*}(u_{q,h},v_{q})+F_{q}^{*}(u_{q,h},v_{q,h})$$

$${}=\dfrac{1}{\Pi^{*}(q)}\left(C(\varphi^{*}(qw_{0}),\psi^{*}(qw_{0}))-C(\varphi^{*}(qw_{0}),\psi^{*}(qw^{1+h/2\kappa^{*}}_{0})\right.$$

$${}-\left.C(\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0}),\psi^{*}(qw_{0}))+C(\varphi^{*}(qw^{1+h/2\kappa^{*}}_{0}),\psi^{*}(qw^{1+h/2\kappa^{*}}_{0}))\right)$$

$${}\geq\dfrac{w^{\kappa^{*}}_{0}}{\ell^{*}(q)}\left(\ell^{*}(qw_{0})-2\ell^{*}(qw^{1+h/4\kappa^{*}}_{0})w^{h/4}_{0}+\ell^{*}(qw^{1+h/2\kappa^{*}}_{0})w^{h/2}_{0}\right)\to w^{\kappa^{*}}_{0}(1-w^{h/4}_{0})^{2}>0$$

when $q\downarrow 0$. In summary, we have proved $\mathbb{P}((\widetilde{U}_{q},\widetilde{V}_{q})\in E_{q,h})>0$ for a sufficiently small $q>0$. This completes the proof of Theorem 3.1.

Fig. 7

The rectangle $E_{q,h}\subset B_{q,h}$ and associated curves.

2.2 A.2. Thresholds and Pseudo Observations

In Figs. 8–10 we depict the differenced log-time-series $x_{t}=\log(x^{0}_{t})-\log(x^{0}_{t-1})$ (left-hand panels) and the extreme pseudo-observations (right-hand panels) that arise from the time series data specified in Section 5. With the thresholds $q\in(0,1)$ reported in Table 6, the time series give rise to paired extreme pseudo-observations that resemble a white noise; see the right-hand panels of Figs. 8–10. To substantiate this claim, we run several portmanteau tests for the null hypothesis

$$\mathcal{H}_{0}:\boldsymbol{\Gamma}_{L}=\mathbf{0},\quad L=1,\ldots,20,$$

where $\boldsymbol{\Gamma}_{L}=\textrm{Cov}(\boldsymbol{\varepsilon}_{t},\boldsymbol{\varepsilon}_{t-L})$ and $\boldsymbol{\varepsilon}_{t}$, $t=1,\dots,m_{q}$, are the residuals obtained by fitting the original data to the time series model VARMA for sufficiently many lags [31]. The selected portmanteau tests include those of Mahdi and McLeod [31], Box and Pierce [8], Ljung and Box [30], Hosking [23], and Li and McLeod [28]. The percentages of $p$-values above the 5$\%$ significance level (meaning that the null of white noise is retained) are given in Table 6, where we also report the sample sizes $m_{q}$.

Fig. 8

Original $x_{t}$’s (a, c, e) and the pairs of extreme pseudo-observations (b, d, f) for foreign currency exchange rates from January 4, 1971, to October 25, 2019.

Fig. 9

Original $x_{t}$’s (a, c, e) and the pairs of extreme pseudo-observations (b, d, f) for stock market indices from January 4, 1971, to February 28, 2020.

Fig. 10

Original $x_{t}$’s (a, c, e) and the pairs of extreme pseudo-observations (b, d, f) for diverse financial instruments from February 5, 1971, to March 3, 2020.

Table 6 The percentages of $p$-values retaining the null of white noise at the significance level $\alpha=0.05$ alongside the sample sizes $m_{q}$ (in parentheses) for appropriate choices of $q$

Table 7 Testing $H_{0}$ vs. $H_{1}$ of pseudo observations of foreign currency exchange rates

Table 8 Testing $H_{0}$ vs. $H_{1}$ of pseudo observations of stock market indices

Table 9 Testing $H_{0}$ vs. $H_{1}$ of pseudo observations of diverse financial instruments

The different choices of $q\in(0,1)$ warrant an explanation. First, we want to work with as small $q>0$ as possible, mainly due to two reasons:

• the estimator’s deterministic bias becomes small (recall Theorem 3.1),

• the time series of extreme pseudo-observations becomes nearly a white-noise.

Working close to a white noise is useful as it helps to reliably calculate critical values of the hypothesis tests for bound (3.13), which we need for the use of Theorem 3.1.

2.3 A.3. Testing the Validity of Bound (3.13)

For the real time series that we are exploring, we want to statistically test the reasonableness of bound (3.13). For this, we adapt the Kolmogorov–Smirnov (K–S), Cramér–von Mises (C–vM), Anderson–Darling (A–D) one-sided statistics [38]:

$$\sqrt{m_{q}}\sup_{(u,v)\in[0,1]\times[0,1]}\{uv-F^{*}_{q,\mathcal{M}_{q}}(u,v)\}_{+},$$

(A.20)

$$m_{q}\int\limits_{[0,1]\times[0,1]}\big{(}\{uv-F^{*}_{q,\mathcal{M}_{q}}(u,v)\}_{+}\big{)}^{2}\textrm{d}F^{*}_{q,\mathcal{M}_{q}}(u,v),$$

(A.21)

$$m_{q}\int\limits_{[0,1]\times[0,1]}\dfrac{\big{(}\{uv-F^{*}_{q,\mathcal{M}_{q}}(u,v)\}_{+}\big{)}^{2}}{u(1-u)v(1-v)}\textrm{d}F^{*}_{q,\mathcal{M}_{q}}(u,v),$$

(A.22)

respectively, where $F^{*}_{q,\mathcal{M}_{q}}$ is defined by Eq. (3.5). Specifically, we use these three statistics to test the null $H_{0}$ of having bound (3.13) versus the alternative $H_{1}$ of not having the bound:

$$H_{0}:\quad\text{$F_{q}^{*}(u,v)\geq uv$ for all $(u,v)\in[0,1]\times[0,1]$}$$

$$H_{1}:\quad\text{$F_{q}^{*}(u,v)<uv$ for some $(u,v)\in[0,1]\times[0,1]$}.$$

Note A.2. The null $H_{0}$ can be reformulated as $F_{q}^{*}\geq C^{\perp}$ and the alternative $H_{1}$ as $F_{q}^{*}\ngeq C^{\perp}$.

The critical values of the tests are obtained by sampling from the pairs of pseudo observations. Namely, we calculate the test statistics, repeat the procedure $N=10\,000$ times, obtain so many values of the test statistics, and finally calculate the 95th percentiles of the respective test-statistic values. The decision rule is to retain the null $H_{0}$ if the test statistic is smaller than the critical value, and to reject it otherwise. The obtained results are summarized in Tables A.2–A.4, where the abbreviations ‘‘stat.,’’ ‘‘crit.,’’ and ‘‘deci.’’ stand for the test statistic value, the critical value, and the decision, respectively. The decision is to retain the null $H_{0}$ when the statistic is smaller than the critical value. In every case, the three tests retain the null $H_{0}$.

To gain an additional insight, in Section A.4 we test the null of the equation $F_{q}^{*}(u,v)=uv$ for all $(u,v)\in[0,1]\times[0,1]$, which is the ‘‘boundary’’ of the null $H_{0}$ introduced earlier.

2.4 A.4. Testing the Boundary Case of Bound (3.13)

In the examples of Section A.3, all of which retained the null $H_{0}$, we now statistically test the reasonableness of the boundary case $F_{q}^{*}(u,v)=uv$ for all $(u,v)\in[0,1]\times[0,1]$. We adapt the Kolmogorov–Smirnov (K–S), Cramér–von Mises (C–vM), Anderson–Darling (A–D) one-sided statistics [cf. 38]

$$\sqrt{m_{q}}\sup_{(u,v)\in[0,1]\times[0,1]}\{F^{*}_{q,\mathcal{M}_{q}}(u,v)-uv\}_{+},$$

(A.23)

$$m_{q}\int\limits_{[0,1]\times[0,1]}\big{(}\{F^{*}_{q,\mathcal{M}_{q}}(u,v)-uv\}_{+}\big{)}^{2}\textrm{d}F^{*}_{q,\mathcal{M}_{q}}(u,v),$$

(A.24)

$$m_{q}\int\limits_{[0,1]\times[0,1]}\dfrac{\big{(}\{F^{*}_{q,\mathcal{M}_{q}}(u,v)-uv\}_{+}\big{)}^{2}}{u(1-u)v(1-v)}\textrm{d}F^{*}_{q,\mathcal{M}_{q}}(u,v),$$

(A.25)

respectively. Specifically, we use them to test the hypotheses

$$H_{0}^{*}:\quad\text{$F_{q}^{*}(u,v)=uv$ for all $(u,v)\in[0,1]\times[0,1]$},$$

$$H_{1}^{*}:\quad\text{$F_{q}^{*}(u,v)>uv$ for some $(u,v)\in[0,1]\times[0,1]$}.$$

Note A.3. The null $H_{0}^{*}$ can be viewed as the ‘‘boundary’’ $F_{q}^{*}=C^{\perp}$ of the null $H_{0}$ introduced in Appendix A.3.

Coming now back to our main discussion, we note that the procedures for calculating the critical values for any of the three tests for $H_{0}^{*}$ vs. $H_{1}^{*}$ using statistics (A.23)–(A.25) are analogous to those we used in Section A.3 for $H_{0}$ vs. $H_{1}$. The decision to retain the null $H_{0}^{*}$ is, of course, taken when the statistic is smaller than the critical value. Our findings are summarized in Tables 10, 11.

Table 10 Testing $H_{0}^{*}$ vs. $H_{1}^{*}$ of pseudo observations of foreign currency exchange rates

Table 11 Testing $H_{0}^{*}$ vs. $H_{1}^{*}$ of pseudo observations of stock market indices

Table 12 Testing $H_{0}^{*}$ vs. $H_{1}^{*}$ of pseudo observations of diverse financial instruments

The null $H_{0}^{*}$ of $F_{q}^{*}=C^{\perp}$ for the pairs (CAD, GBP) and (JPY/USD, NASDAQ) is not rejected by any of the three tests, but the TOMD estimates are equal to 1.5488 and 1.4002, respectively, as seen from Tables 3 and 5. These values suggest that the coordinates of the two aforementioned pairs may actually be fairly dependent.