The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application

Abstract

In this paper, we investigate the CUSUM-type estimator of mean change-point models based on m-asymptotically almost negatively associated (m-AANA) sequences. The family of m-AANA sequences contains AANA, NA, m-NA, and independent sequences as special cases. Under some weak conditions, some convergence rates are obtained such asOP(n1/p−1),OP(n1/p−1log1/pn)andOP(nα−1), where0≤α<1and1<p≤2. Our rates are better than the ones obtained by Kokoszka and Leipus (Stat. Probab. Lett., 1998, 40, 385–393). In order to illustrate our results, we do perform simulations based on m-AANA sequences. As important applications, we use the CUSUM-type estimator to do the change-point analysis based on three real data such as Quebec temperature, Nile flow, and stock returns for Tesla. Some potential applications to change-point models in finance and economics are also discussed in this paper.

1. Introduction

Change-point problems originally arose in the context of quality control, where one typically observes the output of a production line and would wish to signal deviation from an acceptable level while observing the data. The change-point problems may be changes of the mean, variance, and other parameters. Therefore, detecting a change-point and estimating its location are both very important in data processing, modeling, estimation, and inference. The cumulative sum (CUSUM) method is a popular method to solve this problem; see Csörgő and Horváth [1] and Shiryaev [2], and among others. In this paper, we consider the mean change-point models, which can be applied in many fields. For example, investors pay attention to the mean changes of the economic growth rate, consumption level, exchange rate, stock returns, and so on. Therefore, we consider the mean change-point models in this paper. For some0<^τ*<1, let^k*=⌊n^τ*⌋. Here,⌊x⌋denotes the largest integer not exceeding x. Forn≥1, suppose the observations_X1,…,_Xnsatisfy the model:

_Xi=_θ0+_δnI(^k*+1≤i≤n)+_Zi,1≤i≤n,

where mean parameter_θ0, change-amount_δn, as well as change-point location^k*are unknown and_Z1,…,_Zn are mean zero random variables. In Model (1), the estimators of^k*and^τ* based on the CUSUM method (see Kokoszka and Leipus [3]) are, respectively:

_k^n(α)=argmax1≤k≤n−1|_Uk(α)|and_τ^n(α)=_k^n(α)n,

where:

_Uk(α)=^{k(n−k)n1−α}1k∑i=1k_Xi−1n−k∑i=k+1n_Xi,1≤k≤n−1,

and0≤α<1 . Kokoszka and Leipus [3] used the Hájek–Rényi-type inequalities to obtain the convergence rate of CUSUM-type estimator_τ^n(α). In this paper, we also study the consistency of estimator_τ^n(α) in (2) based on dependent sequences of{_Zn,n≥1} . Now, let us recall some related definitions. Block et al. [4] introduced an important concept of negative associated (NA) random variables, which can be applied in reliability theory, percolation theory, and multivariate analysis.

Definition 1.

A finite family of random variables{_Zi,1≤i≤n}is said to be NA if for every pair of disjoint subsets A and B of 1, 2,…, n,

Cov(f(_Zi,i∈A),g(_Zj,j∈B))≤0.

whenever f and g are coordinatewise nondecreasing and the covariance exists.

Motivated by the notion of NA random variables, Chandra and Ghosal [5] introduced the concept of asymptotically almost negatively associated (AANA) random variables.

Definition 2.

A sequence{_Zn,n≥1}of random variables is called AANA if there exists a nonnegative sequenceq(n)→0asn→∞such that:

Cov(f(_Zn),g(_Zn+1,…,_Zn+k))≤q(n)^{[Var(f(_Zn))Var(g(_Zn+1,…,_Zn+k))]1/2},

for alln≥1,k≥1and for all coordinatewise nondecreasing continuous functions f and g whenever the variances exist. The sequence{q(n),n≥1}is said to be the mixing coefficients of{_Zn,n≥1}.

Hu et al. [6] gave a natural extension of m-NA from NA random variables.

Definition 3.

Letm≥1be a fixed integer. A sequence of random variables{_Zn,n≥1}is said to be m-NA if for anyn≥2and any_i1,_i2,…,_in, such that|_ik−_ij|≥mfor all1≤k≠j≤n, we have that_Zi1 ,_Zi2 ,…,_Zin are NA random variables.

Motivated by Hu et al. [6], Nam et al. [7] gave the concept of m-AANA.

Definition 4.

Letm≥1be a fixed integer. A sequence of random variables{_Zn,n≥1}is said to be m-AANA if there exists a nonnegative sequenceq(n)→0asn→∞such that:

Cov(f(_Zn),g(_Zn+m,…,_Zn+k))≤q(n)^{[Var(f(_Zn))Var(g(_Zn+m,…,_Zn+k))]1/2},

for alln≥1,k≥m, and for all coordinatewise nondecreasing continuous functions f and g whenever the variances exist.

Nam et al. [7] obtained the maximal inequalities for m-AANA sequences and gave its applications to Hájek–Rényi-type inequalities and the strong law of large numbers. Ko [8] extended the results of Nam et al. [7] to the Hilbert space. The family of m-AANA sequences contains AANA (withm=1 ), NA, m-NA, and independent sequences as special cases. The notions of NA and AANA have received increasing attention recently. One can refer to [9,10,11,12,13,14,15], etc.

For the mean change-point model (1), Shi et al. [16] extended the results of Kokoszka and Leipus [3] to NA sequences and obtained the strong convergence rate for the estimator_τ^n(α) in (2). Since the m-AANA sequence is weaker than the NA sequence, we study the convergence rate for the estimator_τ^n(α) based on m-AANA sequences. For more research on the change-point models, we can refer to many works such as [1,17,18,19,20,21,22,23,24] and the references therein. In addition, many researchers have joined the study of change-point models in mathematical finance and econometrics. For example, Shiryaev [25] considered a Brownian motion with mean drift, depending onθ,

_Xt=μ(t−θ)I(t≥θ)+σ_Bt,t≥0,

whereμ≠0andσ>0are known constants (as a rule) andθis the “disorder” time or change-point location, which can be either a random variable or simply an unknown parameter. Here,{_Bt,t≥0} is a standard Brownian motion. Obviously, (4) is very important to study Black–Scholes models in finance and economics. For more stochastic models of asset pricing in finance based on change-points, we can refer to Shiryaev [25].

The rest of this paper is organized as follows. Section 2 presents some convergence rates of_τ^n(α)−^τ*(for example,_OP(^n1/p−1),_OP(^{n1/p−1 log1/p}n)and_OP(^nα−1) ). Section 3 provides some simulations to check the results obtained in this paper. As important applications, three real data examples are provided to do the mean change-point analysis in Section 4. The conclusions and further research are discussed in Section 5. Lastly, the main proofs are presented in Section 6.

2. Main Results First, some assumptions are listed as follows.

Assumption 1.

For some1<p≤2, let{_Zn,n≥1}be a mean zero sequence of m-AANA random variables withsupn≥1E^|_Zn|p<∞and the mixing coefficient sequence{q(n),n≥1}satisfy_∑n=1∞ ^q2(n)<∞.

Assumption 2.

For0≤α<1and1<p≤2, denote:

_gn(α,p)=^n1/p−1,if0≤α<1p,^{n1/p−1 log1/p}n,ifα=1p,^nα−1,if1p<α<1.

Let:

_δn≠0and_gn(α,p)/_δn→0asn→∞.

Then, we obtain the convergence rate of_τ^n(α)−^τ*in Theorem 1.

Theorem 1.

Let1<p≤2and Assumptions 1 and 2 be satisfied. Then, for any given0≤α<1,

_τ^n(α)−^τ*=_OP(_gn(α,p)/_δn).

As an application of Theorem 1, we have Corollary 1.

Corollary 1.

Let1<p≤2. If_δn=_δ0≠0in Theorem 1, then for any given0≤α<1,

_τ^n(α)−^τ*=_OP(^n1/p−1),if0≤α<1p,_OP(^{n1/p−1 log1/p}n),ifα=1p,_OP(^nα−1),if1p<α<1.

Remark 1.

In the mean change-point model (1), if_δn=_δ0≠0and{_Zn,n≥1}is an independent and identically distributed sequence of random variables withE_Z1=0andVar(_Z1)=^σ2>0, then, by Corollary 1, it has:

_τ^n(α)−^τ*=_OP(^n−1/2),if0≤α<12,_OP(^{n−1/2 log1/2}n),ifα=12,_OP(^nα−1),if12<α<1,

which was obtained by (1.5) of Kokoszka and Leipus [3]. In addition, Kokoszka and Leipus [3] considered the sequence{_Zn,n≥1}satisfying the following dependent condition:

Var∑i=jm_Zi≤C^(m−j+1)β,

uniformly inj,m,1≤j≤m≤n, where0≤β<2. Denote:

_g˜n(α,β)=^{n−(1/2−β/4)},if0≤α<1/2−β/4,^{n−(1/2−β/4) log1/2}n,ifα=1/2−β/4,^nα−1,if1/2−β/4<α<1.

Let:

_δn≠0and_g˜n(α,β)/_δn→0asn→∞.

Then, Kokoszka and Leipus [3] obtained that:

_τ^n(α)−^τ*=_OP(_g˜n(α,β)/_δn),

(see Corollary 1.1 of [3]). Obviously, our rate (6) is better than that of (9). Thus, our Theorem 1 extends Theorem 1.1 and Corollary 1.1 of [3] to the case of the m-AANA sequence. On the other hand, one can take_δn→0or_δn→∞ in (6) (or (9)), provided_gn(α,p)/_δn→0(or_g˜n(α,β)/_δn→0) asn→∞. Furthermore, let_δn=_δ0≠0and{_Zn,n≥1} be an NA sequence. Shi et al. [16] obtained a strong convergence rate_τ^n−^τ*=o(M(n)n), a.s. for anyM(n)satisfyingM(n)→∞andM(n)/n=o(1) . Our convergence rates are weaker than Shi et al. [16]; however, our m-AANA sequence{_Zn,n≥1}is weaker than the NA sequence, and our change-amount_δncan go to zero or infinity.

3. Simulations

In the mean change-point model (1), we assume that there exists a mean change point location^k*such that:

_Xi=_θ0+_δnI(^k*+1≤i≤n)+_Zi,1≤i≤n,

where_Z1,…,_Znsatisfy:

(_Z1,_Z2,…,_Zn)=d_{w1 Nn}(0,_In)+_{w2 Nn}(0,_Σn),

where_w1,_w2≥0,_w1+_w2=1,_Inis an identity matrix, and_Σnis:

_Σn=_{1+1/nρ^ρ2⋯^ρn−1ρ1+2/nρ⋯^ρn−2ρ2ρ1+3/n⋯^ρn−3⋯⋯⋯⋯⋯^ρn−1⋯^ρ2ρ2n×n}

and|ρ|<1. It is easy to verify that{_Z1,…,_Zn}is a m-AANA sequence withm=2and mixing coefficientsq(n)=O(|ρ^|n). For simplicity, we take_θ0=1,^τ*=0.5,^k*=⌊^τ*n⌋,_w1=_w2=1/2, andρ=−0.6 in (10)–(12) to do the simulation with 10000 replications. For the samplen=50,100,200,800,1400,2000 , Figure 1 shows the box plots of_τ^n(α)−^τ*with differentα(for example,α=0,0.1,0.5,0.7,0.9) and_δn(for example,_δn=^n−0.3,^n−0.2,^n−0.1,ⁿ⁰,^n0.1), where_τ^n(α) is defined by (2).

In Figure 1, the y-axis is the value of_τ^n(α)−^τ* , and the x-axis is the sample n. By the box plots in Figure 1, the differences of_τ^n(α)−^τ* go to zero as sample n increases, which agrees with the consistency of (6) in Theorem 1. It has a similar performance if we take different valuesρ, so the details are omitted here.

4. Real Data Examples

In this section, we use the CUSUM-type estimator_k^n(α)=n_τ^n(α) in (2) to do the mean change-point analysis with three real datasets. The first dataset is for the monthly mean temperature of Quebec in Canada from 1944 to 2008. The data can be found at http://climate.weather.gc.ca. For simplicity, we take the data of monthly mean temperatures for June and October, which contain 76 observations denoted by_x1,iand_x2,i,1≤i≤76, respectively. Let_xi=_x1,iifi=1,…,76and_xi=_x2,i−76ifi=76+1,…,152 . Figure 2 shows the plot graph of_xi,1≤i≤152, where the y-axis is the value of temperature_xiand the x-axis is the sample n.

Obviously, the mean temperature of June is different from that of October, so the change-point location^k*is 76 (or^τ*=0.5). Now, we use the CUSUM-type estimator_k^n(α)to detect^k*, i.e.,_k^n(α)is defined by:

_k^n(α)=argmax1≤k≤n−1|_Uk(α)|

where0≤α<1and:

_Uk(α)=^{k(n−k)n1−α}1k∑i=1k_xi−1n−k∑i=k+1n_xi,1≤k≤n−1.

Table 1 shows the values of_k^n(α)with different valuesαsuch asα=0,0.1,…,0.9.

By Table 1, the estimator_k^n(α)with differentαsuccessfully detects the true change-point location^k*=76.

Second, we also use estimator_k^n(α) to detect a time series of the annual flow of the Nile River at Aswan from 1871 to 1970 (see, for example, Zeileis et al. [26]). It measures annual discharge at Aswan in¹⁰⁸ m³ and is depicted in Figure 3 (there are 100 observations denoted by_xi,1≤i≤100 ). In Figure 3, the y-axis is the annual flow of the Nile River_xi, and the x-axis is the sample n.

Similar to Table 1, the values of_k^n(α) are given in Table 2.

By Table 2, we get a change-point location of 28 or equivalently the year 1898 (see Figure 3). On the other hand, Zeileis et al. [26] and Gao et al. [27] respectively used F-statistics and CUSUM statistics to detected the same change-point location of 28. It is well known that Aswan dam was built in 1898. It significantly changes the annual flow of the river Nile.

Lastly, we do the change-point analysis of returns based on a financial time series. Let_Ptbe the closing prices of Tesla stock. Therefore, the return is defined as_rt=log_Pt−log_Pt−1 . Figure 4 shows 138 daily returns on the prices of Tesla stock from 1 August 2016 to 13 February 2017. In Figure 4, the y-axis is the i-th return_ri, and the x-axis is the sample n. The data were downloaded from Yahoo Finance. Tesla announced on 22 November 2016 that it had completed the acquisition of SolarCity. It seams that the mean returns changed after that time of 22 November 2016 (the observation is 80). Therefore, we perform the test for this change-point of mean returns.

Similar to Kokoszka and Leipus’s CUSUM estimator_k^n(α) defined by (2), for any given0≤α<1 , Antoch et al. [17] investigated the following CUSUM estimator_k˜nof^k*defined as:

_k˜n(α)=argmax1≤k≤n−1|_U˜k(α)|,

where:

_U˜k(α)=^nk(n−k)α∑i=1k_xi−_x¯n,1≤k≤n−1,

and_x¯n=1n_{∑i=1n xi}. Therefore, we use these CUSUM-type estimators_k^n(α) by (13) and_k˜n(α) by (14) to detect the change-point location^k*, where_xi=_ri,1≤i≤138. With the differentα=0,0.1,…,0.9, the values of_k^n(α)and_k˜n(α) are presented in Table 3.

By Table 3, the estimators_k^n(α)and_k˜n(α) find the same change-point location of 86 (1 December 2016). Thus, the capital market recognized Tesla’s acquisition of SolarCity on 22 November of 2016, and the mean returns significantly changed from negative to positive after the time of 1 December 2016 (see Figure 4).

5. Conclusions

The CUSUM method is a popular method to detect the change-point. In this paper, we investigate the consistency of CUSUM-type estimator_τ^n(α)based on m-AANA sequences, which contain many dependent sequences such as NA, m-NA, and AANA sequences. Under the p-th moment (1<p≤2) condition, we obtain a general consistency rate_τ^n(α)−^τ*=_OP(_gn(α,p)/_δn), where_gn(α,p) is defined by (6) and0≤α<1. By taking_δn=_δ0≠0in Theorem 1, we obtain the convergence rates as:

_τ^n(α)−^τ*=_OP(^n1/p−1),if0≤α<1p,_OP(^{n1/p−1 log1/p}n),ifα=1p,_OP(^nα−1),if1p<α<1.

Therefore, our Theorem 1 and Corollary 1 generalize the results of Kokoszka and Leipus [3]. In addition,_δnin Theorem 1 can be taken as_δn→0or_δn→∞, if_gn(α,p)/_δn→0asn→∞. Let_δn=_δ0≠0and{M(n),n≥1}be any positive constant sequence satisfyingM(n)→∞andM(n)/n=o(1) . Shi et al. [16] obtained a strong convergence rate_τ^n−^τ*=o(M(n)n) , a.s. for the case of NA sequence. Our convergence rates are weaker than Shi et al. [16]; however, the m-AANA sequence is weaker than the NA sequence, and the change-amount_δn can go to zero or infinity. In order to check our results, some simulations are shown in Figure 1, which agree with the consistency of Theorem 1. Lastly, three real dataset of Quebec temperature, Nile flow, and returns for Tesla in Section 4 are discussed to show that the CUSUM-type estimator_k^n(α) defined by (13) can successfully detect the change-point location. In addition, it is interesting for scholars to study the strong convergence rate and limit distribution of CUSUM-type estimator_τ^n(α) based on the m-AANA sequence or other dependent sequences in future research. Furthermore, Shiryaev [25] discussed the stochastic disorder problems, which are known as the quickest detection problems. For example, Shiryaev [25] considered the Black–Scholes models with mean drift. Thus, we should pay attention to the applications of change-point models in mathematical finance and econometrics.

6. Proofs of the Main Results

For convenience, in the proofs, letC,_C1,_C2,…be some positive constants that are independent of n and may have different values in different expressions.

Lemma 1.

(Theorem 3 of Nam et al. [7]). For some1<p≤2, let{_Zn,n≥1}be an m-AANA sequence of zero mean random variables with mixing coefficients{q(n),n≥1}satisfying_∑n=1∞ ^q2(n)<∞. Let{_bn,n≥1}be a nondecreasing sequence of positive numbers. Then, for anyε>0and any integern≥1, we have:

Pmax1≤k≤n|1_bk∑j=1k_Zj|≥ε≤^{2p mp−1} _Cp^εp∑j=1nE|_Zj ^|p_bjp,

where_Cpis a positive constant depending only on p.

Proof of Theorem 1.

Let_τn=⌊k/n⌋. For any given0≤α<1 , we have by (1) and (3) that:

E_Uk(α)=^{k(n−k)n1−α}1k∑i=1kE_Xi−1n−k∑i=k+1nE_Xi=−_δn ^n1−α _τn1−α ^{(1−_τn)−α}(1−^τ*),ifk≤^k*,−_δn ^{n1−α (1−_τn)1−α} _τn−α ^τ*,ifk>^k*,

and:

E_U^k* (α)=−_δn ^{n1−α (τ*)1−α (1−τ*)1−α}.

Therefore, we have by (3.11) of Kokoszka and Leipus [3] that:

|_δn|τ¯^n1−α|_τ^n−^τ*|≤2max1≤k≤n−1|_Uk(α)−E_Uk(α)|,

whereτ¯:=(1−α)^{(τ*)−α (1−τ*)−α}min{^τ*,1−^τ*} . By (1) and (3), it is easy to see that:

^nα−1max1≤k≤n−1|_Uk(α)−E_Uk(α)|≤^nα−1max1≤k≤n−11^kα|∑i=1k(_Xi−E_Xi)|+^nα−1max1≤k≤n−11^(n−k)α|∑i=k+1n(_Xi−E_Xi)|=^nα−1max1≤k≤n−11^kα|∑i=1k_Zi|+^nα−1max1≤k≤n−11^(n−k)α|∑i=k+1n_Zi|:=_D1+_D2.

By (5), (17), and (18), in order to prove (6), we have to prove that:

_Di=_OP(_gn(α,p)),i=1,2,

where_gn(α,p) is defined (5). On the one hand, for anyε>0, it follows from Lemma 1 withsupn≥1E^|_Zn|p<∞that:

Pmax1≤k≤n−11^kα|∑i=1k_Zi|>ε^n1−α≤_C1 ^(εn1−α)p∑i=1nE|_Zi ^|piαp≤_C2^{εp np−αp}∑i=1nE|_Zi ^|piαp≤_C3 ^{ε−p n1−p}if0≤α<1p,_C4 ^{ε−p n1−p}lognifα=1p,_C5 ^{ε−p np(α−1)}if1p<α<1,

where_Ci,i=1,…,5 , are positive constants independentof n. Consequently, by (18)–(20), we have:

_D1=_OP(_gn(α,p)).

On the other hand, it can be checked that:

_D2=^nα−1max1≤k≤n−11^(n−k)α|∑i=k+1n_Zi|=^nα−1max1≤k<n1^kα|∑i=1k_Zn−i+1|.

Therefore, one can obtain analogously the same result that

_D2=_OP(_gn(α,p)).

Thus, the proof of (6) is completed.  □

Proof of Corollary 1.

By taking_δn=_δ0≠0in Theorem 1, one can immediately obtain Corollary 1.  □

Figure 1. The box plots ofτ^n(α)-τ*with differentαandδn.

Figure 1. The box plots ofτ^n(α)-τ*with differentαandδn.

Figure 2. The plot graph of monthly mean temperatures based on June and October data.

Figure 3. The plot graph of the annual flow of the Nile River from 1871 to 1970.

Figure 4. The plot graph of the mean returns of Tesla stock from 2016 to 2017.

α	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
k^n(α)	76	76	76	76	76	76	76	76	76	76

α	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
k^n(α)	28	28	28	28	28	28	28	28	28	28

α	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
k^n(α)	86	86	86	86	86	86	86	86	86	86
k˜n(α)	86	86	86	86	86	86	86	86	86	86

Author Contributions

Supervision W.Y.; software S.D. and X.D.; writing, original draft preparation, S.D., X.L., and W.Y. All authors read and agreed to the published version of the manuscript.

Funding

This work is supported by NNSF of China (11701004, 11801003), NSF of Anhui Province (2008085MA14, 1808085QA03, 1808085QA17), and Provincial Natural Science Research Project of Anhui Colleges (KJ2019A0006).

Acknowledgments

The authors are deeply grateful to the Editors and anonymous referees for their careful reading and insightful comments. The comments led to the significant improvement of the paper.

Conflicts of Interest

The authors declare no conflict of interest.

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