Change-Point Detection Under Dependence Based on Two-Sample U-Statistics

Abstract

We study the detection of change-points in time series. The classical CUSUM statistic for detection of jumps in the mean is known to be sensitive to outliers. We thus propose a robust test based on the Wilcoxon two-sample test statistic. The asymptotic distribution of this test can be derived from a functional central limit theorem for two-sample U-statistics. We extend a theorem of Csörgő and Horváth to the case of dependent data.

Appendix: Some Auxiliary Results from the Literature

In this section, we collect some known lemmas and theorems for weakly dependent data. We start with some results on the behaviour of partials sums:

Lemma 3 (Borovkova, Burton, Dehling [3], Lemma 2.23).

Let $(X_{k})_{k\in \mathbb{Z}}$ be a 1-approximating functional with constants (a_k)_k≥0 of an absolutely regular process with mixing coefficients (β(k))_k≥0. Suppose moreover that EX_i = 0 and that one of the following two conditions holds:

• 1. X ₀ is bounded a.s. and $\sum _{k=0}^{\infty }(a_{k} +\beta (k)) <\infty.$

• 2. $E\vert X_{0}\vert ^{2+\delta } <\infty$ and $\sum _{k=0}^{\infty }(a_{k}^{ \frac{\delta }{ 1+\delta } } +\beta ^{ \frac{\delta }{1+\delta } }(k)) <\infty.$

Then, as $N \rightarrow \infty$ ,

$$\displaystyle{ \frac{1} {N}ES_{N}^{2} \rightarrow EX_{ 0}^{2} + 2\sum _{ j=1}^{\infty }E(X_{ 0}X_{j}) }$$

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and the sum on the r.h.s. converges absolutely.

Lemma 4 (Borovkova, Burton, Dehling [3], Lemma 2.24).

Let $(X_{k})_{k\in \mathbb{Z}}$ be a 1-approximating functional with constants (a _k ) of an absolutely regular process with mixing coefficients (β(k)) _k≥0 . Suppose moreover that EX _i = 0 and that one of the following two conditions holds:

• 1. X ₀ is bounded a.s. and $\sum _{k=0}^{\infty }k^{2}(a_{k} +\beta (k)) <\infty.$

• 2. $E\vert X_{0}\vert ^{4+\delta } <\infty$ and $\sum _{k=0}^{\infty }k^{2}(a_{k}^{ \frac{\delta }{ 3+\delta } } +\beta ^{ \frac{\delta }{4+\delta } }(k)) <\infty.$

Then there exits a constant C such that

$$\displaystyle{ ES_{N}^{4} \leq CN^{2}. }$$

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Theorem 4 (Borovkova, Burton, Dehling [3], Theorem 4).

Let $(X_{k})_{k\in \mathbb{Z}}$ be a 1-approximating functional with constants (a _k ) _k≥0 of an absolutely regular process with mixing coefficients (β(k)) _k≥0 . Suppose moreover that EX _i = 0, $E\vert X_{0}\vert ^{4+\delta } <\infty$ and that

$$\displaystyle{ \sum _{k=0}^{\infty }k^{2}(a_{ k}^{ \frac{\delta }{ 3+\delta } } +\beta ^{ \frac{\delta }{4+\delta } }(k)) <\infty, }$$

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for some δ > 0. Then, as $n \rightarrow \infty,$

$$\displaystyle{ \frac{1} {\sqrt{n}}\sum _{i=1}^{n}X_{ i} \rightarrow \mathcal{N}(0,\sigma ^{2}), }$$

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where $\sigma ^{2} = EX_{0}^{2} + 2\sum _{j=1}^{\infty }E(X_{0}X_{j}).$ In case $\sigma ^{2} = 0$ , $\mathcal{N}(0,0)$ denotes the point mass at the origin. If X ₀ is bounded, the CLT continues to hold if  (77) is replaced by the condition that $\sum _{k=0}^{\infty }k^{2}(a_{k} +\beta (k)) <\infty$ .

An important tool to derive asymptotic results for weakly dependent data are coupling methods. We will apply this method in the proof of Proposition 2.

Theorem 5 (Borovkova, Burton, Dehling [3], Theorem 3).

Let $(X_{n})_{n\in \mathbb{N}}$ be a 1-approximating functional with summable constants (a _k ) _k≥0 of an absolutely regular process with mixing rate (β(k)) _k≥0 .  Then given integers K,L and N, we can approximate the sequence of $(K + 2L,N)-$ blocks (B _s ) _s≥1 by a sequence of independent blocks $(B_{s}^{{\prime}})_{s\geq 1}$ with the same marginal distribution in such a way that

$$\displaystyle{ P(\vert \vert B_{s} - B_{s}^{{\prime}}\vert \vert \leq 2\alpha _{ L}) \geq 1 -\beta (K) - 2\alpha _{L}, }$$

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where $\alpha _{L}:= \left (2\sum _{l=L}^{\infty }a_{l}\right )^{1/2}.$

In statistical application, the question of how to estimate $\sigma ^{2}$ is important. In the situation when the observations are a functional of α-mixing process, Dehling et al. [10] propose the estimation of the variance of partial sums of dependent processes by the subsampling estimator

$$\displaystyle{ \hat{D}_{n} = \frac{1} {[n/l_{n}]}\sqrt{ \frac{\pi } {2}}\sum _{i=1}^{[n/l_{n}]}\frac{\vert \hat{T}_{i}(l_{n}) - l_{n}\tilde{U}_{n}\vert } {\sqrt{l_{n}}} }$$

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with $\hat{T}_{i}(l) =\sum _{ j=(i-1)l+1}^{il}F_{n}(X_{j})$ and $\tilde{U}_{n} = \frac{1} {n}\sum _{j=1}^{n}F_{ n}(X_{j})$, where F _n (⋅ ) is the empirical distribution function.

Theorem 6 (Dehling, Fried, Sharipov, Vogel, Wornowizki [9], Theorem 1.2).

Let (X _k ) _k≥1 be a stationary, 1-approximating functional of an α-mixing processes. Suppose that for some δ > 0, $E\vert X_{1}\vert ^{2+\delta } <\infty$ , and that the mixing coefficients (α _k ) _k≥1 and the approximation constants (a _k ) _k≥1 satisfy

$$\displaystyle{ \sum _{k=1}^{\infty }(\alpha _{ k})^{ \frac{2} {2+\delta } } <\infty,\quad \sum _{k=1}^{\infty }(a_{k})^{\frac{1+\delta } {2+\delta } } <\infty. }$$

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In addition, we assume that F is Lipschitz-continuous, that $\alpha _{k} = O(n^{-8})$ and that $a_{m} = O(m^{-12})$ . Then, as $n \rightarrow \infty$ , $l_{n} \rightarrow \infty$ and $l_{n} = o(\sqrt{n})$ , we have $\hat{D}_{n}\longrightarrow \sigma$ in L ₂ .

To deal with the degenerate kernel g, we need to find upper bounds for the expectations $E\left (g(X_{i_{1}},X_{j_{1}})g(X_{i_{2}},X_{j_{2}})\right )$, in terms of the maximal distance among the indices. Since 1 ≤ i ₁ < i ₂ ≤ [n λ] and $[n\lambda ] + 1 \leq j_{1} <j_{2} \leq n$, we get i ₁ < i ₂ < j ₁ < j ₂.

Lemma 5 (Dehling, Fried [8], Proposition 6.1).

Let (X _n ) _n≥1 be a 1-approximating functional with constants (a _k ) _k≥1 of an absolutely regular process with mixing coefficients (β(k)) _k≥1 and let g(x,y) be a 1-continuous bounded degenerate kernel. Then we have

$$\displaystyle{ \vert E(g(X_{i_{1}},X_{j_{1}})g(X_{i_{2}},X_{j_{2}}))\vert \leq 4S\phi (a_{[k/3]}) + 8S^{2}(\sqrt{a_{ [k/3]}} +\beta ([k/3])) }$$

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where $S = \vert \sup _{x,y}g(x,y)\vert$ and $k =\max \left \{i_{2} - i_{1},j_{1} - i_{2},j_{2} - j_{1}\right \}$ .

The following two results are useful for proving tightness of a stochastic process. The first one is used to control the fluctuation of maximum. Let ξ ₁, …, ξ _n be random variables, and define $S_{k} =\xi _{1} +\ldots +\xi _{k}$ (S ₀ = 0), and $M_{n} =\max _{0\leq k\leq n}\vert S_{k}\vert$.

Theorem 7 (Billingsley [2], Theorem 10.2).

Suppose that β ≥ 0 and α > 1∕2 and that there exist nonnegative numbers u ₁ ,…,u _n such that for all positive λ

$$\displaystyle{ P\left (\vert S_{j} - S_{i}\vert \geq \lambda \right ) \leq \frac{1} {\lambda ^{4\beta }} \left (\sum _{i<l\leq j}u_{l}\right )^{2\alpha },\quad 0 \leq i \leq j \leq n\quad, }$$

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then for all positive λ

$$\displaystyle{ P\left (M_{n} \geq \lambda \right ) \leq \frac{K_{\beta,\alpha }} {\lambda ^{4\beta }} \left (\sum _{0<l\leq n}u_{l}\right )^{2\alpha }, }$$

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where K _β,α is a constant depending only on β and α.

Theorem 8 (Billingsley [2], Theorem 8.4).

The sequence {Y _n }, defined by

$$\displaystyle{ Y _{n}(t) = \frac{1} {\sigma \sqrt{n}}S_{[nt]} + (nt - [nt]) \frac{1} {\sigma \sqrt{n}}\xi _{[nt]+1} }$$

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is tight if for each ε > 0 there exist a λ > 1 and a $n_{0} \in \mathbb{N}$ such that for n ≥ n ₀

$$\displaystyle{ P\left (\max _{i\leq n}\vert S_{k+i} - S_{k}\vert \geq \lambda \sigma \sqrt{n}\right ) \leq \frac{\epsilon } {\lambda ^{2}}. }$$

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