Robust Change Point Test for General Integer-Valued Time Series Models Based on Density Power Divergence

In this study, we consider the problem of testing for a parameter change in general integer-valued time series models whose conditional distribution belongs to the one-parameter exponential family when the data are contaminated by outliers. In particular, we use a robust change point test based on density power divergence (DPD) as the objective function of the minimum density power divergence estimator (MDPDE). The results show that under regularity conditions, the limiting null distribution of the DPD-based test is a function of a Brownian bridge. Monte Carlo simulations are conducted to evaluate the performance of the proposed test and show that the test inherits the robust properties of the MDPDE and DPD. Lastly, we demonstrate the proposed test using a real data analysis of the return times of extreme events related to Goldman Sachs Group stock.


Introduction
Integer-valued time series models have received widespread attention from researchers and practitioners in diverse research areas. Since the works of McKenzie [1] as well as Al-Osh and Alzaid [2], integer-valued autoregressive (INAR) models have gained popularity in the analysis of correlated time series of counts. Later, as an alternative, Ferland et al. [3] proposed using Poisson integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models (see Engle [4] and Bollerslev [5]). Since then, INGARCH models have been studied by many authors, such as Fokianos et al. [6], who developed Poisson autoregressive (Poisson AR) models with nonlinear specifications for their intensity processes. The Poisson assumption on INGARCH models has been extended to include negative binomial INGARCH (NB-INGARCH) models (Davis and Wu [7] and Christou and Fokianos [8]), zero-inflated generalized Poisson INGARCH models (Zhu [9,10] and Lee et al. [11]), and one-parameter exponential distribution AR models (Davis and Liu [12]). The latter are also known as general integer-valued time series models and have been studied by, among others, Diop and Kengne [13] and Lee and Lee [14], who considered change point tests for these models.
The change point problem is a core issue in time series analysis because changes can occur in underlying model parameters owing to critical events or policy changes, and ignoring such changes can result in false conclusions. Numerous studies exist on change point analysis in time series models; refer to Kang and Lee [15] and Lee and Lee [14], and the articles cited therein, for the background and history of change points in integer-valued time series models. Lee and Lee [14] conducted a comparison study of the performance of various cumulative sum (CUSUM) tests using score vectors and residuals through the Monte Carlo simulations. In their work, the conditional maximum likelihood where F t−1 is a σ-field generated by Y t−1 , Y t−2 , . . . and f θ (x, y) is a non-negative bivariate function defined on [0, ∞) × N 0 , N 0 = N ∪ {0}, depending on the parameter θ ∈ Θ ⊂ R d , and satisfies inf θ∈Θ f θ (x, y) ≥ x * for some x * > 0 for all x, y. Here, p(·|·) is a probability mass function, given by p(y|η) = exp{ηy − A(η)}h(y), y ≥ 0, where η is the natural parameter and A(η) and h(y) are known functions. This distribution family includes several famous discrete distributions, such as the Poisson, negative binomial, and binomial distributions. If B(η) = A (η), B(η t ) and B (η t ) become the conditional mean and variance of Y t , and X t = B(η t ). The derivative of A(η) exists for the exponential family; see Lehmann and Casella [29]. Since B (η t ) = Var(Y t |F t−1 ) > 0, B(η) is strictly increasing, and since B(η t ) = E(Y t |F t−1 ) > 0, A(η) is also strictly increasing. To emphasize the role of θ, we also use X t (θ) and η t (θ) = B −1 (X t (θ)) to stand for X t and η t , respectively.
Davis and Liu [12] showed that the assumption below ensures the strict stationarity and ergodicity of {(X t , Y t )}: They also demonstrated that there exists a measurable function f θ . .) almost surely (a.s.). Meanwhile, the DPD d α between two density functions g and h is defined as For a parametric family {G θ , θ ∈ Θ} with densities given by {g θ } and a distribution H with density h, the minimum DPD functional Then, given a random sample Y 1 , . . . , Y n with unknown density h, the MDPDE is defined byθ where L α,n (θ) = 1 n ∑ n t=1 l α,t (θ) and When α = 0 and 1, the MDPDE becomes the MLE and the L 2 -distance estimator, respectively. Basu et al. [16] revealed thatθ α,n is consistent for T α (H) and asymptotically normal. Furthermore, the estimator is robust against outliers, but still exhibits high efficiency when the true distribution belongs to a parametric family {G θ } and α is close to zero. The tuning parameter α controls the trade-off between robustness and asymptotic efficiency. A large α escalates the robustness while a small α yields greater efficiency. The conditional version of the MDPDE is defined similarly (cf. Section 2 of Kim and Lee [22]). For Y 1 , . . . , Y n generated from (1), the MDPDE for general integer-valued time series models is defined asθ andη t (θ) = B −1 ( X t (θ)) is updated recursively using the following equations: with an arbitrarily chosen initial value X 1 . The MDPDE with α = 0 becomes the CMLE from (3). Kim and Lee [22] showed that under the regularity conditions (A0)-(A9) stated below, the MDPDE is strongly consistent and asymptotically normal. Conditions (A10) and (A11) are imposed to derive the limiting null distribution of the DPD-based change point test in Section 2.2. Below, V and ρ ∈ (0, 1) represent a generic integrable random variable and a constant, respectively; the symbol · denotes the L 2 -norm for matrices and vectors; and E(·) is taken under θ 0 , where θ 0 denotes the true value of θ.

Remark 2.
Instead of (A6), Kim and Lee [22] assumed to prove Proposition 1. Note that this condition is satisfied directly if (A3) and (A6) hold. In our study, we alter the above condition to (A6) to prove Lemma A1 in the Appendix A, which is needed to obtain the limiting null distribution of the DPD-based change point test in Section 2.2.
The following INGARCH(1,1) models are typical examples of general integer-valued time series models: Condition (A0) trivially holds, and the process {(X t , Y t ), t ≥ 1} has a strictly stationary and ergodic solution. Condition (A1) can be replaced with the following: (A1) The true parameter θ 0 lies in a compact neighborhood Θ ∈ R 3 + of θ 0 , where Moreover, we can express where the initial value X 1 is taken as d/(1 − a) for simplicity. Based on the above and (A4), the conditions (A2), (A5), and (A7)-(A10) are all satisfied for INGARCH(1,1) models, as proven by Theorem 3 of Kang and Lee [15]. Kim and Lee [22] showed recently that the following Poisson and negative binomial INGARCH(1,1) models satisfy (A3) and (A4). Furthermore, following the arguments presented in Section 3.2 of their study, (A6) holds for these models as well. Below, we show that (A11) holds for Poisson and negative binomial INGARCH(1,1) models.

DPD-Based Change Point Test
As a robust test for parameter changes in general integer-valued time series models, we propose a DPD-based test for the following hypotheses: To construct the test, we employ the objective function of the MDPDE. That is, our test is constructed using the empirical version of the DPD. Let L α,n be that in (2). To implement our test, we employ the following test statistic: is a consistent estimator of K α . For the consistency of K α , see Lemma A5 in Appendix A.
Using the mean value theorem (MVT), we have the following, for each s ∈ [0, 1], where θ * α,n,s is an intermediate point betweenθ α,n and θ 0 . From ∂ L α,n (θ α,n )/∂θ = 0, we obtain that, for s = 1, Furthermore, since J α is nonsingular (cf. proof of Lemma 7 in Kim and Lee [22]), this can be expressed as Substituting the above into (4) yields In Appendix A, we show that the first two terms on the right-hand side of (5) converge weakly to standard Brownian bridge and the last term is asymptotically negligible. Therefore, we obtain the following theorem.
We reject H 0 if T α n is large; see Table 1 of Lee et al. [32] for the critical values. When a change point is detected, its location is estimated as

Remark 3.
The proposed test T α n with α = 0 is the same as the score-vector-based CUSUM test proposed by Lee and Lee [14], given by (3),θ 0,n is the CMLE, and I n = n −1 ∑ n t=1 ∂ 2l 0,t (θ 0,n )/∂θ∂θ T . In the next section, we compare the performance of T α n with that of T score n in the presence of outliers.

Simulation
In this section, we evaluate the performance of the proposed test T α n (with α > 0) through simulations, focusing on the comparison with T score n . First, we consider the Poisson INGARCH models: where X 1 is set to 0 for the data generation and X 1 is set as the sample mean of the data. The sample sizes considered are n = 500 and 1000, with 1000 repetitions for each simulation.  and T α n ) exhibit reasonable size, even when a + b is close to 1. When n = 500, T score n outperforms T α n in terms of power; however, as the sample size increases to n = 1000, T α n exhibits similar power to that of T score n , particularly when α is small. The power of T α n tends to decrease as α increases, confirming that an MDPDE with large α results in a loss of efficiency.   To evaluate the robustness of the proposed test, we assume that contaminated data Y c,t are observed instead of Y t in (6) (cf. Fried et al. [33]): where P t are independent and identically distributed (iid) Bernoulli random variables with success probability p and Y o,t are iid Poisson random variables with mean γ. We assume that Y t , P t , and Y o,t are all independent. In this simulation, we consider the cases p = 0.01, 0.03 and γ = 5, 10. The results are reported in Tables 2-5, showing that T score n suffers from size distortions that become more severe as either p or γ increase. In contrast, T α n compensates for this defect remarkably well, yielding comparable power to that of T score n when n = 1000. This indicates that as more data are contaminated by outliers, T α n increasingly outperforms T score n .      Next, we consider the following NB-INGARCH(1,1) models: where X 1 and X 1 are 0 and the sample mean of the data, respectively. We set r = 10, and use the same parameter settings as in the Poisson INGARCH model case. In order to evaluate the robustness of the test, we observe contaminated data Y c,t , as in (7), where Y t are generated from (8), P t are iid Bernoulli random variables with success probability p, and Y o,t are iid NB(10, κ) random variables. We consider the cases p = 0.01, 0.03 and κ = 0.6, 0.5. The results are reported in Tables 6-10, showing similar results to those in Tables 1-5. Our findings show that the DPD-based test performs reasonably well in terms of both size and power, regardless of the existence of outliers. In addition, we confirm that the proposed test outperforms the score-based CUSUM test when the data are contaminated by outliers.     Table 9. Empirical sizes and powers for NB-INGARCH(1,1) models when p = 0.03 and κ = 0.6.

Real Data Analysis
In this section, we demonstrate the validity of T α n using a real data analysis. To this end, we analyze the return times of extreme events related to GS stock, which are constructed based on the daily log-returns for the period of 5 May 1999 to 15 March 2012. Davis and Liu [12] and Kim and Lee [22] previously investigated this data set in their works on geometric INGARCH(1,1) models (i.e., NB-INGARCH(1,1) models with r = 1).
We first compute the hitting times, τ 1 , τ 2 , . . ., for which the log-returns of GS stock fall outside the 0.05 and 0.95 quantiles of the data. The return times of these extreme events are calculated as Y t = τ t − τ t−1 . Since Y t ≥ 1, we consider a geometric distribution that counts the total number of trials, rather than the number of failures, to fit the following geometric INGARCH(1,1) models to the data: where X 1 is set as the sample mean of the data. Kim and Lee [22] showed that the optimal α for the MDPDE is 0.25, using the criterion provided in Remark 1. The results for the parameter estimation are summarized in Table 11 for α = 0 (CMLE) and 0.25 (MDPDE with optimal α); figures in parentheses denote the standard errors of the corresponding estimates. We observe that, compared with the CMLE, the MDPDE with α = 0.25 is quite different and has smaller standard errors. Next, we use T score n and T 0.25 n ( T α n with α = 0.25) to perform a parameter change test at the nominal level of 0.05 (the corresponding critical value is 3.004). Let T score n = max 1≤k≤n SCORE k,n and T 0.25 n = max 1≤k≤n DPD k,n . The left and right panels of Figure 2 display SCORE k,n and DPD k,n , respectively. For most k, DPD k,n appears to be smaller than SCORE k,n , which is definitely attributed to the robustness of the MDPDE and DPD. We obtain T score n = 5.136, which suggests the existence of a parameter change. In Figures 1 and 2, the red, vertical, dashed line represents the location of a change when T score n is applied. However, this result is not so reliable because T score n can signal a change point affected by outliers as seen in the previous section, and the change point is truly detected at the occurrence time of an outlier in this case. In contrast, T 0.25 n yields a value of 1.219, indicating that no change point exists. This result clearly demonstrates that outliers can severely affect parameter estimates and change point tests by mistakenly identifying a change point. Our findings confirm that the DPD-based change point test provides a functional and robust alternative to the score-based CUSUM test in the presence of outliers.

Conclusions
In this study, we developed a DPD-based robust change point test for general integer-valued time series models with a conditional distribution that belongs to the one-parameter exponential family. We provided regularity conditions under which the proposed test converges weakly to the function of a Brownian bridge. The simulation study showed that the DPD-based test produces reasonable sizes and powers regardless of the existence of outliers, whereas the score-based CUSUM test suffers from severe size distortions when the data are contaminated by outliers. In the real data analysis using the return times of extreme events related to GS stock, the score-based CUSUM test supported the existence a parameter change, due to the influence of outliers, while the DPD-based test did not detect a change point because of its robust property. This result confirms the validity of the proposed test as a robust test in practice. It is noteworthy that the DPD-based test can be feasibly extended to other parametric models as far as the asymptotic properties of the MDPDE for the models are validated. We leave the issue of extension to other models as our future study.

Conflicts of Interest:
The authors declare no conflict of interest.