key: cord-0849690-y53fl7md
authors: Ling, Jin; Li, Xiao-qin; Yang, Wen-zhi; Jiao, Jian-ling
title: The CUSUM statistic of change point under NA sequences
date: 2021-12-22
journal: Appl Math
DOI: 10.1007/s11766-021-4015-z
sha: 3321257f7cfeb67c815e2c3cb95952df6d1b71c1
doc_id: 849690
cord_uid: y53fl7md

In this paper, we investigate the CUSUM statistic of change point under the negatively associated (NA) sequences. By establishing the consistency estimators for mean and covariance functions respectively, the limit distribution of the CUSUM statistic is proved to be a standard Brownian bridge, which extends the results obtained under the case of an independent normal sample and the moving average processes. Finally, the finite sample properties of the CUSUM statistic are given to show the efficiency of the method by simulation studies and an application on a real data analysis.

Recall that a finite family {Z 1 , Z 2 , · · · , Z n } is said to be NA if for any disjoint subsets A, B of {1, 2, · · · , n}, and any real coordinatewise nondecreasing functions f on R A , g on R B , Cov(f (Z k , k ∈ A), g(Z k , k ∈ B)) ≤ 0.

A sequence of random variables {Z n , n ≥ 1} is said to be NA if for every n ≥ 2, Z 1 , Z 2 , · · · , Z n are NA.

In this paper, we investigate the asymptotic property of CUSUM statistic of change point under NA sequences. For convenience, let ⌊x⌋ denote the largest integer not exceeding x, and {B 0 (t); t ∈ [0, 1]} be a standard Brownian bridge. Let d → mean the convergence in distribution. Inclán and Tiao [12] proposed a CUSUM statistic to test a change-point of variance as follows: Theorem 1.1 Let {X n , n ≥ 1} be a sequence of independent, identically distributed Normal random variables with X 1 ∼ N (0, σ 2 ) and σ 2 > 0. Then for k = ⌊tn⌋ and 0 ≤ t ≤ 1,

where IT n,k = √ n 2 (

|IT n,k | indicates the existence of a variance change, and the changepoint is at arg max 1≤k≤n |IT n,k |. Meanwhile, Lee and Park [16] extended (1) to an infinite order moving average processes. For more details about the change-point detection, we can refer to the books [5, 7] .

In view of nonnegative of X 2 i in (1), we will further investigate the asymptotic distribution of IT n in (1) based on the nonnegative sequences of NA random variables. By establishing the consistency estimators for mean and covariance functions, the limit distribution of CUSUM statistic of change point is proved to be a standard Brownian bridge. The paper is organized as follows. In Section 2, we give some assumptions and main results of this paper. In Section 3, some simulation studies and a real data analysis are implemented to show the efficiency of the CUSUM statistic. Finally, the proofs of main results are presented in Section 4. §2 Some assumptions and main results Let {Z n , n ≥ 1} be a sequence of strictly stationarity nonnegative NA random variables, and γ(h) be the covariance function of {Z n , n ≥ 1}, which is denoted as γ(h) = Cov(Z 1 , Z 1+h ), for h = 0, 1, 2, . . .. γ(h) is usually unknown and estimated by the sample covariance function

In order to establish the main results, we need the following assumptions.

and 

Then, the estimator for σ 2 0 is given as followŝ

Now, we give the main results of this paper. 

and lim n→∞ E|σ 2 n − σ 2 0 | = 0.

By Theorem 2.1, the limit distribution for the CUSUM statistic is presented as follows. 

whereμ andσ 2 n are defined in (6) and (7), respectively. If t ∈ [0, 1] and k = ⌊nt⌋, then max

as n → ∞. Remark 2.1. Theoretically, it is easy to obtain the consistency of mean estimatorμ (see (6) ) but difficult to establish the consistency ofσ 2 n in (7) based on the auto-covariance function estimatorγ(h). In this paper, we use the truncation method and the covariance inequality of NA sequence (see Lemma 3.1 of Roussas [21] ) to obtain the moment consistency ofσ 2 n in Theorem 2.1. Then, the limit distribution for the CUSUM statistic max 1≤k≤n |T nk | is presented in Theorem 2.2. By (8) , it is easy to establish (1) in Theorem 1.1 obtained by Inclán and Tiao [12] . So, Theorem 2.2 extends the result in the case of normal sequence to the dependent setting of NA sequences. In Section 3, some simulations are carried out to show that the empirical sizes and powers of our CUSUM statistic have a good performance. Further more, we apply our method and the results by Inclán and Tiao [12] to detect a change-point of variances for the returns of log daily prices of Dow Jones Industrial (DJI) index which caused by COVID-19 pandemic in 2020. §3 Simulation studies and a real data analysis

In this subsection, we carry out some simulations to show the empirical sizes and powers for the CUSUM statistic max 1≤k≤n |T nk | in (8) . For convenience, if X and Y have the same

and

where ρ is some constant in (−1, 0]. Let x + = max(x, 0) and x − = max(−x, 0). By Joag-Dev and Proschan [13] , it can be seen that

For simplicity, we do the simulations by 10000 replications and for the case

Let the null hypothesis be H 0 : σ 2 1 = σ 2 2 and the alternative hypothesis be H 1 :

For the significance level α = 0.05, if max 1≤k≤n |T nk | > R * = 1.358, then we reject the null hypothesis and conclude that there is a change-point atk * = arg max 1≤k≤n |T nk | (see Inclán and Tiao [12] ). Consequently, for the significance level α = 0.05, we take h n = ⌊n 1/5 ⌋ in (4) and obtain the empirical sizes and powers for the estimator T n in the following Table 1 . Table 1 . Empirical sizes and powers for CUSUM statistic max 1≤k≤n |T nk |.

By Table 1 , we can see that, the differences of empirical sizes are smaller than 0.05 and the empirical powers go to 1 as the sample size n increasing. Meanwhile, under H 1 : σ 2 1 = 1 and σ 2 2 = 4, we obtain the histograms of estimatork * = arg max 1≤k≤n |T nk | for k * = ⌊ n 2 ⌋, ρ = −n −1.1 and n = 300, 600, 900 in By histograms in Fig 1, the percentage ofk * for k * = ⌊ n 2 ⌋ is increasing as n increasing.

In this subsection, we apply our method and the results by Inclán and Tiao [12] to analysis the average returns of Dow Jones Industrial (DJI) index. Let P t is the price of DJI of day t ∈ T , the return is defined as r t = log P t − log P t−1 . The left of Fig 2 shows Since the method of Inclán and Tiao [12] was only used to test change-point of variance, here we also calculate the value max 1≤k≤n |IT nk | in (1), where X i is replaced by (r i −r). Then max 1≤k≤n |IT nk | = 7.7497 > 1.358 and arg max 1≤k≤n |IT nk | = 204. Both methods have detected the same change-point location at day 204. However, it should be pointed out that our method is not only used to detect the change-point of variance but also can be used to detect the change of nonnegative parameter. On the other hand, we find that the change-point location day 204 (February 20, 2020) is at the early stage of COVID-19 pandemic. As time goes on, the COVID-19 pandemic has caused an obviously catastrophic result to the global economy. Therefore, people all over the world should unite to defeat the COVID-19 pandemic, then humanity will finally overcome this epidemic. §4 Proofs of main results For convenience, let C, C 1 , C 2 , . . . be some positive constants which are independent of n. In addition, ⇒ denotes the weak convergence under the Skorohod topology.

Proof of Theorem 2.1. Obviously, by Lemma 4.2, it is easy to have that

which completes the proof of (6). Next, we will prove (7) . From (3) to (5), it follows

. For x ≥ 0, the function f (x) = x 2 is increasing. Then for the nonnegative of Z i , {Z 2 1 , . . . , Z 2 n } is also a strictly stationarity NA sequence. So by Theorem 2 of Shao [22] with EZ 4 1 < ∞,

Second, we consider L 2 . Sinceγ(h) = 1

, then it can be seen that

Therefore, it follows

Obviously, it follows from Hölder inequality and EZ 4 1 < ∞ that 

Similarly,

Meanwhile, by Hölder inequality, it has

, then it has sup x |f ′ (x)| ≤ 1, a.s.. So, by Lemma 3.1 of Roussas [21] , we obtain that for 1

Then, together with (2), we obtain that

since ρ ∈ (0, 1/4). Consequently, from (16) to (20) , it follows that

Obviously,

By (11), it follows

and

Combining (11) with Theorem 2 of Shao [22] , we get that 

The similarity holds true for

It is easy to have

Therefore, by (15) and (21)-(29), L 2 = o(1). In addition, by (2), as h n → ∞, it can be checked that L 3 = o (1) . Consequently, together with (12) and (13), it has the result of (7).

Proof of Theorem 2.2. Denote S n = ∑ n i=1 (Z i − EZ i ) and X n (t) = (3) and Theorem 4 of Shao [22] , X n (t) ⇒ W (t), so {X n (t) − tX n (1)} ⇒ B 0 (t) (see Page 93 of Billingsley [3] ). Without loss of generality, we assume that k = nt, 1 ≤ k ≤ n, since t ∈ [0, 1] and k = ⌊nt⌋. Then

Therefore, by the fact {X n (t) − tX n (1)} ⇒ B 0 (t), one can obtain that

By (7) |B 0 (t)| is given in equation (9.40) of Billingsley [3] .

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