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Copula-based measures of reflection and permutation asymmetry and statistical tests

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Abstract

We propose measures of copula reflection and permutation asymmetry for data with positive quadrant dependence. We first define the measures of reflection asymmetry using a weighting function and then extend this approach to construct measures of permutation asymmetry for bivariate data. We define the corresponding statistical tests based on these measures and find that the proposed tests have higher statistical power comparing to some other tests for permutation and reflection symmetry studied in the literature. In addition, the measures can be used to summarize dependence structure of a multivariate data set in a few numbers and to select a more appropriate copula in the model.

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Acknowledgments

This research has been supported by an NSERC Discovery Grant. Thanks to Harry Joe and the three referees for valuable comments that helped to improve this paper.

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Correspondence to Pavel Krupskii.

Appendices

Appendix 1: Proof of Proposition 1, Section 3

For \(i = 1,\ldots ,n\), denote \(n_{ji} = 1/2-\sum _{j=1}^d U_{ji}/d\). We have:

It implies that

$$\begin{aligned}&\int _{\sum _{j=1}^d u_j \le d/2} \left( \frac{1}{2}-\frac{1}{d}\sum _{j=1}^d u_j\right) ^k[\widehat{C}_n(u_1,\ldots ,u_d) - (\widehat{C}_R)_n(u_1,\ldots ,u_d)] du_1 \ldots du_d \\&\quad = \widehat{G}(\Delta _R;k), \end{aligned}$$

where

$$\begin{aligned} \widehat{C}_n(u_1,\ldots ,u_d) = \frac{1}{n}\sum _{i=1}^nI(U_{1i} < u_1,\ldots ,U_{di} < u_d) \end{aligned}$$

is the empirical copula. We use the result of Segers (2012) to find that

$$\begin{aligned} \sqrt{n}(\widehat{C}_n - C) \rightarrow _d \mathcal {G}_C, \quad \sqrt{n}((\widehat{C}_R)_n - C_R) \rightarrow _d \mathcal {G}_{C_R}, \end{aligned}$$

where

$$\begin{aligned} \mathcal {G}_{C}(u_1,\ldots ,u_d) = \alpha (u_1,\ldots ,u_d)-\sum _{j=1}^d\frac{\partial C(u_1,\ldots ,u_d)}{\partial u_j}\cdot \alpha (1,\ldots ,1,u_j,1,\ldots ,1), \end{aligned}$$

and \(\alpha \) is a centered Gaussian process with the covariance function

$$\begin{aligned}&\mathrm {Cov}(\alpha (u_1,\ldots ,u_d),\alpha (v_1,\ldots ,v_d))\\&\quad = C(\min \{u_1,v_1\},\ldots ,\min \{u_d,v_d\}) - C(u_1,\ldots ,u_d)C(v_1,\ldots ,v_d). \end{aligned}$$

We get

$$\begin{aligned}&\sqrt{n}(\widehat{G}(\Delta _R;k)- G(\Delta _R;k)) \rightarrow _d \int _{\sum _{j=1}^d u_j \le d/2}\left( \frac{1}{2}-\frac{1}{d}\sum _{j=1}^d u_j\right) ^k\\&\quad \times \,[\mathcal {G}_C(u_1,\ldots ,u_d) - \mathcal {G}_{C_R}(u_1,\ldots ,u_d)] du_1 \ldots du_d, \end{aligned}$$

and the asymptotic normality of \(\widehat{G}(\Delta _R;k)\) follows from Lemma 3.9.8 of van der Vaart and Wellner (1996). \(\square \)

Appendix 2: Proof of Proposition 2, Section 3

Denote \(\tilde{U}_j = 0.5 - U_j\). We have:

$$\begin{aligned} G_3^*= & {} \frac{1}{d^3}\mathrm {E}(\tilde{U}_1+ \cdots +\tilde{U}_d)^3 = \frac{1}{d^3}\sum _{j=1}^d\mathrm {E}\tilde{U}_j^3 + \frac{3}{d^3}\sum _{j_1>j_2}\mathrm {E}\tilde{U}_{j_1}\tilde{U}_{j_2}(\tilde{U}_{j_1} + \tilde{U}_{j_2}) \nonumber \\&+ \frac{6}{d^3}\sum _{j_1>j_2>j_3}\mathrm {E}\tilde{U}_{j_1}\tilde{U}_{j_2}\tilde{U}_{j_3} \nonumber \\= & {} \frac{1}{d^3}\sum _{j_1>j_2}\mathrm {E}(\tilde{U}_{j_1}+\tilde{U}_{j_2})^3 + \frac{6}{d^3}\sum _{j_1>j_2>j_3}\mathrm {E}\tilde{U}_{j_1}\tilde{U}_{j_2}\tilde{U}_{j_3}. \end{aligned}$$
(8)

For simplicity of notation, we use subscripts \((j_1,j_2) = (1,2)\) and \((j_1,j_2,j_3) = (1,2,3)\). Let \(U_1^*, U_2^*, U_3^*\) be iid U(0, 1) random variables independent of \((U_1, U_2, U_3) \sim C_{123}\). Then based on pages 56 and 58 of Joe (2014),

$$\begin{aligned} I_{12}:= & {} \int _0^1\int _0^1u_1u_2c_{12}(u_1,u_2)du_1du_2 = \Pr (U_1^*<U_1, U_2^*<U_2)\\= & {} \int _0^1\int _0^1\overline{C}_{12}(u_1,u_2)du_1du_2 = \int _0^1\int _0^1C_{12}(u_1,u_2)du_1du_2 \end{aligned}$$

with \(\overline{C}_{12}(u_1,u_2) = 1-u_1-u_2+C_{12}(u_1,u_2)\).

$$\begin{aligned} I_{123}:= & {} \int _{[0,1]^3}u_1u_2u_3c_{123}(u_1,u_2,u_3)du_1du_2du_3 = \Pr (U_1^*<U_1, U_2^*<U_2, U_3^*<U_3)\\= & {} \int _{[0,1]^3} \overline{C}_{123}(u_1,u_2,u_3)du_1du_2du_3\\= & {} \int _{[0,1]^3} \left[ 1-u_1-u_2-u_3+C_{12}(u_1,u_2)+C_{13}(u_1,u_3)+C_{23}(u_2,u_3)\right. \\&\left. -C_{123}(u_1,u_2,u_3)\right] du_1du_2du_3\\= & {} -\frac{1}{2} + I_{1,2} + I_{1,3} + I_{2,3} - \int _{[0,1]^3}C_{123}(u_1,u_2,u_3)]du_1du_2du_3. \end{aligned}$$

It implies that

$$\begin{aligned} \mathrm {E}\tilde{U}_{j_1}\tilde{U}_{j_2}\tilde{U}_{j_3}= & {} -I_{j_1j_2j_3} + \frac{1}{2}\left( I_{j_1j_2}+I_{j_1j_3}+I_{j_2j_3}\right) -\\&\frac{1}{4}\left( \int _0^1u_{j_1}du_{j_1} + \int _0^1u_{j_2}du_{j_2} + \int _0^1u_{j_3}du_{j_3}\right) + \frac{1}{8}\\= & {} \frac{1}{4} - \frac{1}{2}(I_{j_1j_2}+I_{j_1j_3}+I_{j_2j_3}) +\\&\int _{[0,1]^3}C_{j_1,j_2,j_3}(u_{j_1},u_{j_2},u_{j_3})du_{j_1}du_{j_2}du_{j_3}. \end{aligned}$$

By reflecting \(U_j\),

$$\begin{aligned} \mathrm {E}\tilde{U}_{j_1}\tilde{U}_{j_2}\tilde{U}_{j_3}= & {} -\frac{1}{4} + \frac{1}{2}(I_{j_1j_2}+I_{j_1j_3}+I_{j_2j_3})\\&\quad - \int _{[0,1]^3}C_{R,j_1,j_2,j_3}(u_{j_1},u_{j_2},u_{j_3})du_{j_1}du_{j_2}du_{j_3}, \end{aligned}$$

and therefore \(\mathrm {E}\tilde{U}_{j_1}\tilde{U}_{j_2}\tilde{U}_{j_3} = \frac{1}{2}\zeta ^*(j_1,j_2,j_3)\), where

$$\begin{aligned} \zeta ^*(j_1,j_2,j_3) := \int _{[0,1]^3}\Delta _R(u_{j_1},u_{j_2},u_{j_3})du_{j_1}du_{j_2}du_{j_3}. \end{aligned}$$

As a result, from (8) we get:

$$\begin{aligned} G_3^* = \frac{3}{d^3}\sum _{j_1>j_2>j_3} \zeta ^*(j_1,j_2,j_3) + O\left( \frac{1}{d}\right) . \end{aligned}$$
(9)

\(\square \)

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Krupskii, P. Copula-based measures of reflection and permutation asymmetry and statistical tests. Stat Papers 58, 1165–1187 (2017). https://doi.org/10.1007/s00362-016-0743-1

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