Consistent nonparametric tests for detecting gradual changes in the marginals and the copula of multivariate time series

Abstract

From a series of observations ${\mathbf Y}_1, \ldots , {\mathbf Y}_n$ in $\mathbb {R}^d$ taken sequentially, an interesting question is to know whether or not a significant change occurred in their stochastic behavior. The problem has been largely investigated both for univariate and multivariate observations, where the null hypothesis states that $F_1 = \cdots = F_n$, where $F_j({\mathbf y}) = \mathrm{P}({\mathbf Y}_j \le {\mathbf y})$. In most of the works done so far, the alternative hypothesis is generally that of an abrupt change at some unknown time K, i.e. $F_j = D_1$ for $j \le K$ and $F_j = D_2$ when $j > K$. This assumption is unrealistic in applications where changes tend to occur gradually. In this paper, a more general gradual-change model is proposed in which one admits the existence of times $K_1 < K_2$ where the distribution smoothly changes from $D_1$ to $D_2$. A general class of consistent test statistics for the detection of gradual changes is introduced and their large-sample behavior is investigated under a general $\alpha$-mixing condition. The proposed framework allows to detect changes in the marginal series as well as in the copula. Monte-Carlo simulations indicate the good sampling properties of the tests and their usefulness is illustrated on climatic data.

Appendices

Appendix A: Proofs

1.1 Proof of Proposition 1

The next lemma will be useful in the sequel; its proof is postponed to “Proof of Lemma 1” in Appendix.

Lemma 1

For any $\mathbf{b} = (b_1, \ldots , b_n) \in \mathbb {R}^n$,

$$\begin{aligned} \langle \varvec{\omega }({\mathbf K}), \mathbf{b} \rangle = { 1 \over K_2-K_1 } \sum _{j=K_1}^{K_2-1} j \left( \bar{b}_{1:j} - \bar{b}_{1:n} \right) , \end{aligned}$$

where $\bar{b}_{1:j}$ is the mean of $b_1, \ldots , b_j$.

From Eq. (6),

$$\begin{aligned} \sqrt{n} \, L_{{\mathbf K},n}({\mathbf y}) = { 1 \over K_2 - K_1 } \sum _{j=K_1}^{K_2-1} \mathbb {F}_n \left( {j\over n}, {\mathbf y}\right) , \end{aligned}$$

where for $(s,{\mathbf y}) \in [0,1] \times \mathbb {R}^d$,

$$\begin{aligned} \mathbb {F}_n(s,{\mathbf y}) = { \lfloor ns \rfloor \over \sqrt{n} } \left\{ F_{1:\lfloor ns \rfloor }({\mathbf y}) - F_{1:n}({\mathbf y}) \right\} \end{aligned}$$

is the sequential empirical process. From Theorem 2 of Philipp and Pinzur (1980),

$$\begin{aligned} \sup _{(s,{\mathbf y}) \in [0,1] \times \mathbb {R}^d} \left| \mathbb {F}_n \left( {\lfloor ns \rfloor \over n}, {\mathbf y}\right) - \mathbb {F}(s,{\mathbf y}) \right| {\mathop {\longrightarrow }\limits ^{\mathrm{P}}}0, \end{aligned}$$

where $\mathbb {F}$ is a tight centered Gaussian process such that $\mathrm{Cov}\left\{ \mathbb {F}(s,{\mathbf y}), \mathbb {F}(s',{\mathbf y}') \right\} = \{ \min (s,s') - ss' \} \, \varGamma ({\mathbf y},{\mathbf y}')$ for each $(s,{\mathbf y}),(s',{\mathbf y}') \in [0,1] \times \mathbb {R}^d$ and

$$\begin{aligned} \varGamma ({\mathbf y},{\mathbf y}') = \sum _{\ell \in \mathbb {Z}} \mathrm{Cov}\left\{ \mathbb {I}\left( {\mathbf Y}_0 \le {\mathbf y}\right) , \mathbb {I}\left( {\mathbf Y}_\ell \le {\mathbf y}' \right) \right\} . \end{aligned}$$

Letting $\varvec{\kappa }= (\kappa _1,\kappa _2) \in (0,1)^2$ be such that ${\mathbf K}= \lfloor n\varvec{\kappa } \rfloor$, one can then write

$$\begin{aligned} \sqrt{n} \, L_{{\mathbf K},n}({\mathbf y}) = { n \over \lfloor n\kappa _2 \rfloor - \lfloor n\kappa _1 \rfloor } \int _{\kappa _1}^{\kappa _2} \mathbb {F}_n \left( { \lfloor ns \rfloor \over n }, {\mathbf y}\right) \mathrm{d}s. \end{aligned}$$

If one defines

$$\begin{aligned} \mathbb {L}(\varvec{\kappa },{\mathbf y}) = \int _{\kappa _1}^{\kappa _2} { \mathbb {F}(s,{\mathbf y}) \over \kappa _2 - \kappa _1 } \, \mathrm{d}s, \end{aligned}$$

then

$$\begin{aligned} \left| \sqrt{n} \, L_{{\mathbf K},n} - \mathbb {L}(\varvec{\kappa },\cdot ) \right|\le & {} { 1 \over \kappa _2-\kappa _1 } \left| \int _{\kappa _1}^{\kappa _2} \left\{ \mathbb {F}_n \left( {\lfloor ns \rfloor \over n}, \cdot \right) - \mathbb {F}(s,\cdot ) \right\} \mathrm{d}s \right| \\&\quad + \, \left| \int _{\kappa _1}^{\kappa _2} \mathbb {F}_n \left( {\lfloor ns \rfloor \over n}, \cdot \right) \mathrm{d}s \right| ~ \left| { n \over \lfloor n\kappa _2 \rfloor - \lfloor n\kappa _1 \rfloor } - { 1 \over \kappa _2 - \kappa _1 } \right| . \end{aligned}$$

The first summand on the righthand side is bounded above by

$$\begin{aligned} \sup _{s \in [0,1]} \left| \mathbb {F}_n \left( {\lfloor ns \rfloor \over n}, \cdot \right) - \mathbb {F}(s,\cdot ) \right| . \end{aligned}$$

Since $\int _{\kappa _1}^{\kappa _2} \mathbb {F}_n(\lfloor ns \rfloor /n,\cdot ) \mathrm{d}s$ is tight, the second summand converges to zero as $n\rightarrow \infty$. As a consequence,

$$\begin{aligned} \sup _{(\varvec{\kappa },{\mathbf y}) \in \Delta \times \mathbb {R}^d} \left| \sqrt{n} \, L_{{\mathbf K},n}({\mathbf y}) - \mathbb {L}(\varvec{\kappa },{\mathbf y}) \right| {\mathop {\longrightarrow }\limits ^{\mathrm{P}}}0. \end{aligned}$$

For the computation of the covariance structure, note that

$$\begin{aligned} \mathrm{Cov}\left\{ \mathbb {L}(\varvec{\kappa },{\mathbf y}), \mathbb {L}(\varvec{\kappa }',{\mathbf y}') \right\}= & {} \mathrm{E}\left\{ \int _{\kappa _1}^{\kappa _2} { \mathbb {F}(s,{\mathbf y}) \over \kappa _2-\kappa _1 } \, \mathrm{d}s \times \int _{\kappa _1'}^{\kappa _2'} { \mathbb {F}(s',{\mathbf y}') \over \kappa _2'-\kappa _1' } \, \mathrm{d}s' \right\} \\= & {} \int _{\kappa _1}^{\kappa _2} \int _{\kappa _1'}^{\kappa _2'} { \mathrm{E}\left\{ \mathbb {F}(s,{\mathbf y}) \times \mathbb {F}(s',{\mathbf y}') \right\} \over (\kappa _2-\kappa _1) (\kappa _2'-\kappa _1') } \, \mathrm{d}s' \, \mathrm{d}s, \end{aligned}$$

where the last equality follows from Fubini’s Theorem. Since from a simple computation, $\mathrm{E}\{ \mathbb {F}(s,{\mathbf y}) \times \mathbb {F}(s',{\mathbf y}') \} = \{ \min (s,s') - s s' \} \, \varGamma ({\mathbf y},{\mathbf y}')$,

$$\begin{aligned} \mathrm{Cov}\left\{ \mathbb {L}(\varvec{\kappa },{\mathbf y}), \mathbb {L}(\varvec{\kappa }',{\mathbf y}') \right\}= & {} \left\{ \int _{\kappa _1}^{\kappa _2} \int _{\kappa _1'}^{\kappa _2'} { \left\{ \min (s,s') - s s' \right\} \over (\kappa _2-\kappa _1) (\kappa _2'-\kappa _1') } \, \mathrm{d}s' \, \mathrm{d}s \right\} \varGamma ({\mathbf y},{\mathbf y}') \\= & {} \varLambda (\varvec{\kappa },\varvec{\kappa }') \, \varGamma ({\mathbf y},{\mathbf y}'). \end{aligned}$$

1.2 Proof of Proposition 2

Since by assumption, $\Psi (rg) = |r| \Psi (g)$, one can write $\sqrt{n} \, \Psi (L_{{\mathbf K},n}) = \Psi ( \sqrt{n} \, L_{{\mathbf K},n} )$. Hence, in view of the conclusion of Proposition 1 and from the Continuous Mapping Theorem, $\sqrt{n} \, \Psi (L_{{\mathbf K},n}) \rightsquigarrow \Psi \left\{ \mathbb {L}(\varvec{\kappa },\cdot ) \right\}$. Then, upon noting that

$$\begin{aligned} S_n^{\Psi } = \int _{[0,1]^2} \mathbb {I}\left( \lfloor n\kappa _1 \rfloor < \lfloor n\kappa _2 \rfloor \right) \eta ( \lfloor n\varvec{\kappa } \rfloor / n) \, \Psi ( \sqrt{n} \, L_{\lfloor n\varvec{\kappa } \rfloor ,n}) \, \mathrm{d}\varvec{\kappa }, \end{aligned}$$

it readily follows that

$$\begin{aligned} S_n^{\Psi } \rightsquigarrow \mathbb {S}^{\Psi } = \int _\Delta \eta (\varvec{\kappa }) \, \Psi \left\{ \mathbb {L}(\varvec{\kappa },\cdot ) \right\} \mathrm{d}\varvec{\kappa }. \end{aligned}$$

Similarly,

$$\begin{aligned} T_n^{\Psi } = \sup _{\varvec{\kappa }\in \Delta } \Psi ( \sqrt{n} \, L_{\lfloor n\varvec{\kappa } \rfloor ,n} ) \rightsquigarrow \mathbb {T}^{\Psi } = \sup _{\varvec{\kappa }\in \Delta } \eta (\varvec{\kappa }) \, \Psi \left\{ \mathbb {L}(\varvec{\kappa },\cdot ) \right\} . \end{aligned}$$

1.3 Proof of Proposition 3

Suppose ${\mathbf Y}_1, \ldots , {\mathbf Y}_n$ have distributions $F_1, \ldots , F_n$ that follow the gradual-change model with distributions $D_1 \ne D_2$ and time of change ${\mathbf G}= (\lfloor n\gamma _1 \rfloor ,\lfloor n\gamma _2 \rfloor ) \in \Delta _n$. Then, for ${\mathbf K}\in \Delta _n$, define the empirical process

$$\begin{aligned} \widetilde{L}_{{\mathbf K},n}({\mathbf y}) = { 1 \over \sqrt{n} } \sum _{j=1}^n \left\{ \omega _j({\mathbf K}) - \bar{\omega }({\mathbf K}) \right\} \left\{ \mathbb {I}({\mathbf Y}_j \le {\mathbf y}) - F_j({\mathbf y}) \right\} . \end{aligned}$$

Using arguments similar as those in the proof of Proposition 1, one can show that $\widetilde{L}_{{\mathbf K},n}({\mathbf y})$ converges weakly uniformly in $\Delta \times \mathbb {R}^d$ to some centered Gaussian process, say $\widetilde{\mathbb {L}}(\varvec{\kappa },{\mathbf y})$. For $\mathbf {F} = (F_1, \ldots , F_n)$, one can write

$$\begin{aligned} L_{{\mathbf K},n}({\mathbf y}) = { \widetilde{L}_{{\mathbf K},n}({\mathbf y}) \over \sqrt{n} } + { \langle \varvec{\omega }({\mathbf K}), \mathbf {F}({\mathbf y}) \rangle \over n } \, , \end{aligned}$$

so that in view of (7),

$$\begin{aligned} L_{{\mathbf K},n}({\mathbf y}) {\mathop {\longrightarrow }\limits ^{\mathrm{P}}}\lim _{n\rightarrow \infty } { \langle \varvec{\omega }({\mathbf K}), \mathbf {F}({\mathbf y}) \rangle \over n } = \left\{ D_1({\mathbf y}) - D_2({\mathbf y}) \right\} \varLambda (\varvec{\kappa },\varvec{\gamma }). \end{aligned}$$

As a consequence, $\Psi (L_{{\mathbf K},n}) {\mathop {\longrightarrow }\limits ^{\mathrm{P}}}\Psi (D_1-D_2) \, \varLambda (\varvec{\kappa },\varvec{\gamma }) > 0$ as $n \rightarrow \infty$ for each ${\mathbf K}\in \Delta _n$. Therefore, $\sqrt{n} \, \Psi (L_{{\mathbf K},n}) \longrightarrow + \infty$ in probability and thus the tests based on $S_n^{\Psi }$ and $T_n^{\Psi }$ are consistent.

1.4 Proof of Proposition 4

From the proof of Proposition 3 and the fact that $\Psi (D_1-D_2) > 0$,

$$\begin{aligned} \varvec{\gamma }_n^{\Psi } \overset{\mathrm{P}}{\longrightarrow }\underset{ \varvec{\kappa }\in \Delta }{\mathrm {argsup}} \, { \Psi (D_1-D_2) \, \varLambda (\varvec{\kappa },\varvec{\gamma }) \over \{ \varLambda (\varvec{\kappa },\varvec{\kappa }) \}^{1/2} } = \underset{ \varvec{\kappa }\in \Delta }{\mathrm {argsup}} \, { \varLambda (\varvec{\kappa },\varvec{\gamma }) \over \{ \varLambda (\varvec{\kappa },\varvec{\kappa }) \}^{1/2} } \, . \end{aligned}$$

Since $\varLambda (\varvec{\kappa },\varvec{\gamma })$ is the limit as $n \rightarrow \infty$ of $\langle \varvec{\omega }({\mathbf K}), \varvec{\omega }({\mathbf G}) \rangle / n$, the Cauchy–Schwartz inequality entails $\varLambda (\varvec{\kappa },\varvec{\gamma }) / \{ \varLambda (\varvec{\kappa },\varvec{\kappa }) \}^{1/2} \le \varLambda (\varvec{\gamma },\varvec{\gamma }) \}^{1/2}$, with equality if and only if $\varvec{\kappa }= \varvec{\gamma }$. One concludes that $\varvec{\gamma }_n^{\Psi } \overset{\mathrm{P}}{\longrightarrow }\varvec{\gamma }$, because

$$\begin{aligned} \underset{ \varvec{\kappa }\in \Delta }{\mathrm {argsup}} \, { \varLambda (\varvec{\kappa },\varvec{\gamma }) \over \{ \varLambda (\varvec{\kappa },\varvec{\kappa }) \}^{1/2} } = \varvec{\gamma }. \end{aligned}$$

1.5 Proof of Proposition 5

From Bücher and Ruppert (2013), $(\mathbb {F}_n,{\widehat{\mathbb {F}}}_n)$ converges weakly to $(\mathbb {F},{\widetilde{\mathbb {F}}})$, where ${\widetilde{\mathbb {F}}}$ is an independent copy of $\mathbb {F}$. Then, note that for $\kappa _1$, $\kappa _2$ such that $K_1 = \lfloor n\kappa _1 \rfloor$ and $K_2 = \lfloor n\kappa _2 \rfloor$,

$$\begin{aligned} {\widehat{L}}_{{\mathbf K},n}({\mathbf y}) = {n \over K_2-K_1 } \int _{\kappa _1}^{\kappa _2} {\widehat{\mathbb {F}}}_n \left( { \lfloor ns \rfloor \over n}, {\mathbf y}\right) \mathrm{d}s. \end{aligned}$$

From arguments similar as those in the proof of Proposition 1, one can conclude that ${\widehat{L}}_{{\mathbf K},n}$ is an independent copy of

$$\begin{aligned} \mathbb {L}_{\mathbf K}({\mathbf y}) = {1 \over \kappa _2 - \kappa _1 } \int _{\kappa _1}^{\kappa _2} \mathbb {F}(s,{\mathbf y}) \, \mathrm{d}s. \end{aligned}$$

1.6 Proof of Lemma 1

First note that for all $j \in \{ 1, \ldots , n \}$,

$$\begin{aligned} \omega _j({\mathbf K}) = \left\{ K_2 - \max (j, K_1) \over K_2 - K_1 \right\} \mathbb {I}(j<K_2). \end{aligned}$$

The remaining of the proof is a computation:

$$\begin{aligned} \langle \varvec{\omega }({\mathbf K}), \mathbf{b} \rangle= & {} \sum _{j=1}^n \left\{ \omega _j({\mathbf K}) - \bar{\omega }({\mathbf K}) \right\} \left( b_j - \bar{b}_{1:n} \right) \\= & {} \sum _{j=1}^n \left( b_j - \bar{b}_{1:n} \right) \omega _j({\mathbf K}) \\= & {} \sum _{j=1}^n \left( b_j - \bar{b}_{1:n} \right) \left\{ K_2 - \max (j, K_1) \over K_2 - K_1 \right\} \mathbb {I}(j<K_2) \\= & {} { 1 \over K_2-K_1 } \sum _{j=1}^n \sum _{i=\max (j,K_1)}^{K_2-1} \left( b_j - \bar{b}_{1:n} \right) \mathbb {I}(j<K_2) \\= & {} { 1 \over K_2-K_1 } \sum _{j=1}^n \sum _{i=1}^n \left( b_j - \bar{b}_{1:n} \right) \mathbb {I}\left\{ \max (j,K_1) \le i< K_2 \right\} \mathbb {I}(j<K_2) \\= & {} { 1 \over K_2-K_1 } \sum _{i=K_1}^{K_2-1} \sum _{j \le i} \left( b_j - \bar{b}_{1:n} \right) \\= & {} { 1 \over K_2-K_1 } \sum _{j=K_1}^{K_2-1} j \left( \bar{b}_{1:j} - \bar{b}_{1:n} \right) . \end{aligned}$$

Appendix B: Explicit expressions

1.1 Test statistics

First define $A, Z, Z^\star \in \mathbb {R}^{n\times n}$ such that

$$\begin{aligned} A_{ji} = \mathbb {I}({\mathbf Y}_j \le {\mathbf Y}_i), \quad Z_{\ell \cdot } = \sum _{j \le \ell } A_{j\cdot } \quad \text{ and } \quad Z^\star _{\ell \cdot } = \sum _{j \le \ell } j \, A_{j}. \end{aligned}$$

Then, observe that

$$\begin{aligned} W_{\mathbf K}= \left( \sqrt{n} \, L_{{\mathbf K},n}({\mathbf Y}_1), \ldots , \sqrt{n} \, L_{{\mathbf K},n}({\mathbf Y}_n) \right) = { \varvec{\omega }({\mathbf K}) \, A - \bar{\omega }({\mathbf K}) \, \mathbf{1} \, A \over \sqrt{n} } \,. \end{aligned}$$

One has $\bar{\omega }({\mathbf K}) \, \mathbf{1} \, A = \bar{\omega }({\mathbf K}) \, Z_{n\cdot }$, where $\bar{\omega }({\mathbf K}) = (K_1+K_2-1) / 2n$ and from the definition of $\varvec{\omega }({\mathbf K})$ and straightforward computations, one obtains

$$\begin{aligned} \varvec{\omega }({\mathbf K}) \, A = \sum _{j=1}^n \omega _j({\mathbf K}) \, A_{j\cdot } = \left( K_2 \, Z_{K_2-1,\cdot } - K_1 \, Z_{K_1,\cdot } \over K_2 - K_1 \right) - \left( Z^\star _{K_2-1,\cdot } - Z^\star _{K_1,\cdot } \over K_2 - K_1 \right) . \end{aligned}$$

The computation of Z and $Z^\star$ can exploit the fact that $Z_{1\cdot } = Z^\star _{1\cdot } = A_{1\cdot }$ and for $\ell \in \{ 2, \ldots , n \}$, $Z_{\ell \cdot } = Z_{\ell -1,\cdot } + A_{\ell \cdot }$ and $Z^\star _{\ell \cdot } = Z^\star _{\ell -1,\cdot } + \ell \, A_{\ell \cdot }$. Finally,

$$\begin{aligned} \Psi ^\mathrm{CvM}(\sqrt{n} \, L_{{\mathbf K},n})= & {} \sqrt{ {1 \over n } \sum _{i=1}^n W_{{\mathbf K},i}^2 }, \\ \Psi ^\mathrm{KS}(\sqrt{n} \, L_{{\mathbf K},n})= & {} \max _{1 \le i \le n} \left| W_{{\mathbf K},i} \right| , \\ \Psi ^\mathrm{Kui}(\sqrt{n} \, L_{{\mathbf K},n})= & {} \max _{1 \le i \le n} W_{{\mathbf K},i} - \min _{1 \le i \le n} W_{{\mathbf K},i}. \end{aligned}$$

1.2 Multiplier versions

In order to derive expressions for the multiplier versions of the test statistics, let $A^{(j)} \in \mathbb {R}^{j \times n}$ be the matrix of the first j lines of A and define

$$\begin{aligned} \varvec{\gamma }^{(j)} = \left( { \xi _1 \over \bar{\xi }_{1:j} } - 1, \ldots , { \xi _j \over \bar{\xi }_{1:j} } - 1 \right) . \end{aligned}$$

One can then write

$$\begin{aligned} \left( {\widehat{\mathbb {F}}}_n \left( { j \over n }, {\mathbf Y}_1 \right) , \ldots , {\widehat{\mathbb {F}}}_n \left( { j \over n }, {\mathbf Y}_n \right) \right) = { 1 \over \sqrt{n} } \left( \varvec{\gamma }^{(j)} A^{(j)} - { j \over n } \, \varvec{\gamma }^{(n)} A \right) . \end{aligned}$$

Let $\widehat{Z}, \widehat{Z}^\star \in \mathbb {R}^{n\times n}$ be such that $\widehat{Z}_{\ell \cdot } = \varvec{\gamma }^{(\ell )} A^{(\ell )}$ and $\widehat{Z}^\star _{\ell \cdot } = \sum _{j\le \ell } \varvec{\gamma }^{(j)} A^{(j)}$. With this notation,

$$\begin{aligned} \widehat{W}_{\mathbf K}= \left( {\widehat{L}}_{{\mathbf K},n}({\mathbf Y}_1), \ldots , {\widehat{L}}_{{\mathbf K},n}({\mathbf Y}_n) \right)= & {} { 1 \over \sqrt{n} (K_2-K_1) } \left\{ \widehat{Z}^\star _{K_2 \cdot } - \widehat{Z}^\star _{K_1 \cdot }\right. \\&\quad \left. -\,\left( { 1 \over n } \sum _{j=K_1+1}^{K_2} j \right) \widehat{Z}_{n\cdot } \right\} . \end{aligned}$$

Observe that $\widehat{Z}_{1\cdot } = \widehat{Z}^\star _{1\cdot } = \varvec{\gamma }^{(1)} A^{(1)}$ and for $\ell \in \{ 2, \ldots , n \}$, $\widehat{Z}_{\ell \cdot } = \widehat{Z}_{\ell -1,\cdot } + \varvec{\gamma }^{(\ell )}_\ell \, A_{\ell \cdot }$ and $\widehat{Z}^\star _{\ell \cdot } = \widehat{Z}^\star _{\ell -1,\cdot } + \widehat{Z}_{\ell \cdot }$. Finally,

$$\begin{aligned} \Psi ^\mathrm{CvM}({\widehat{L}}_{{\mathbf K},n})= & {} \sqrt{ {1 \over n } \sum _{i=1}^n \widehat{W}_{{\mathbf K},i}^2 }, \\ \Psi ^\mathrm{KS}({\widehat{L}}_{{\mathbf K},n})= & {} \max _{1 \le i \le n} \left| \widehat{W}_{{\mathbf K},i} \right| , \\ \Psi ^\mathrm{Kui}({\widehat{L}}_{{\mathbf K},n})= & {} \max _{1 \le i \le n} \widehat{W}_{{\mathbf K},i} - \min _{1 \le i \le n} \widehat{W}_{{\mathbf K},i}. \end{aligned}$$