Testing the epidemic change in nearly nonstationary autoregressive processes

Consider random variables X1, X2, . . . , Xn with parameters of interest θ1, . . . , θn, n > 2. Testing an epidemic type change (or changed segment) in these parameters means testing null hypothesis θ1 = θ2 = · · · = θn against the alternative θ1 = · · · = θk∗ = θm∗+1 = · · · = θn = μ0 and θk∗+1 = · · · = θm∗ = μ1 for some unknown 1 < k∗ < m∗ 6 n and μ0 6= μ1. Here k∗ is the beginning, m∗ is the end and `∗ = m∗ − k∗ is the length or duration of the epidemic state. To the best of our knowledge such a problem for independent observations have been formulated for the first time by Levin and Kline [1] (we also refer to [2, Sect. 1.4]). Yao [3] have studied various test statistics in order to detect an epidemic change in the mean values of a sequence of independent normally distributed random variables. Ramanayake and Gupta [4, 5] investigated various likelihood ratio type statistics and independent random variables from an exponential family. Graiche et al. [6] investigated the changed segment problem for α-mixing random variables. For more information on this subject, see [7, Sects. 9.3, 9.4] and [8–11].

To the best of our knowledge such a problem for independent observations have been formulated for the first time by Levin and Kline [1] (we also refer to [2,Sect. 1.4]). Yao [3] have studied various test statistics in order to detect an epidemic change in the mean values of a sequence of independent normally distributed random variables. Ramanayake and Gupta [4,5] investigated various likelihood ratio type statistics and independent random variables from an exponential family. Graiche et al. [6] investigated the changed segment problem for α-mixing random variables. For more information on this subject, see [7,Sects. 9.3,9.4] and [8][9][10][11].
Assume we are given an n-sample y n,1 , . . . , y n,n generated by y n,k = φ n y n,k−1 + k + a n,k , k = 1, . . . , n, n 1, y n,0 = 0, where the parameter φ n ∈ (0, 1) satisfies φ n → 1 as n → ∞, ( k , k 1) are i.i.d. centered, at least square integrable random variables, (a n,k ) is a sequence that will be precised later. The process (1), when φ n → 1 as n → ∞ is a nearly nonstationary first order autoregressive process with drift. Throughout the paper, the parameter φ n is supposed to be known. Our aim is to propose tests for the null hypothesis (H 0 ) a n,1 = · · · = a n,n = 0 against the epidemic or changed segment alternative (H A ) there exist 1 k * n < m * n n such that a n,k = a n 1 I * n (k), a n = 0, 1 k n, where I * n is the epidemics interval I * n = {k * n + 1, . . . , m * n } and 1 I * n denotes its indicator function.
So, due to (4), we have the epidemic change model, where a sequence of dependent random variables satisfying the null hypothesis is shifted by a deterministic sequence. This is the reason why statistics (2) seems very natural in this situation.
We study limit behavior of T α,n for α = 0 (Levin and Kline statistic) and α ∈ (0, 1/2 − 1/p), p > 2, (Račkauskas and Suquet statistics) trying to see how the use of extra weighting improves the detection of (relatively) short epidemics. Of course the range of detection will be smaller here than that in the case of independent samples. If α = 0, then the innovations are required to have finite second moment. For another case, the innovations should satisfy the stronger integrability condition In this paper, we study two types of models depending on parameterization of the coefficient φ n in (1). The first type model corresponds to φ n = e γ/n , γ < 0, see [14]. The second type model corresponds to φ n = 1 − γ n n , where γ n → ∞, and γ n n → 0 as n → ∞, see [15]. As we shall see, the limit behavior of T α,n differs for these two types of models. The paper is organized as follows. Section 2 is devoted to the limit behaviour of statistics under null hypothesis. In Section 3, we show the consistency of test statistics T α,n . We investigate the power of the test in Section 4. Final section is devoted to some auxiliary results.

Limit behavior of test statistics under null hypothesis
The next two processes play the central role in this paper. We denote by W = {W (t), 0 t 1} the standard Wiener process and by U γ = {U γ (t), t ∈ [0, 1]} the following Ornstein-Uhlenbeck process: As usual, C[0, 1] is the Banach space of continuous functions with uniform norm where The functional (9) appears in the limit of our test statistics. Throughout the paper E → denotes convergence in distribution in the metric space E as n → ∞. Accordingly, the classical convergence in distribution of a sequence of random variables is denoted by R → as n → ∞, while convergence in probability is denoted by P → as n → ∞.

Levin and Kline statistic
We start with Levin and Kline statistic T 0,n . First, we study the model (1) with the coefficient φ n = e γ/n , γ < 0. Under the assumption of square integrability of innovations, we obtain that the limit of such statistic is a functional of an integrated Ornstein-Uhlenbeck process.
Theorem 1. Under (H 0 ), for the first type model defined by (1) and (6), where Proof. Consider the functionals g n and g defined on the continuous function space where www.mii.lt/NA By the special case of Lemma A.1 where α = 0, the functionals g n and g are Lipschitz where (S pl n (t), t ∈ [0, 1]) is the polygonal line partial sums process build on the observations (y n,k−1 ): From Theorem 1 in [16], Note that limit theorems in [16] are proved with σ 2 = 1 for simplicity, but the results hold the same for σ 2 = 1 as well. Lemma A.1 now gives and the convergence (10) follows from (12), (14), (15) and the continuous mapping theorem.
Now we find the limit of test statistic T 0,n under null hypothesis in second type model.

Theorem 2.
Under (H 0 ), for the second type model defined by (1) and (7), where σ 2 = E 2 1 . Proof. The proof of this theorem is essentially the same as the proof of Theorem 1 using Theorem 2 in [16] instead of Theorem 1 in [16] and Lemma A.1 given below.

T α,n statistics with α > 0
Now we show that, for the model (1) with φ n = e γ/n , γ < 0, the limit of T α,n (α > 0) is a functional of an integrated Ornstein-Uhlenbeck process. Here we need a stronger integrability on innovations than just a second moment.
Further we find the limit of test statistics T α,n under null hypothesis in the second type model, i.e., where coefficient φ n in model (1) is defined by φ n = 1 − γ n /n, γ n → ∞ and γ n /n → 0 as n → ∞. The limit of these statistics is a functional depending on Wiener process. Here the requirement is not only integrability condition on innovations, but also the rate of divergence of γ n . Theorem 4. In the second type model defined by (1) and (7), assume that ( i ) satisfy condition (5) for some p > 2. Then, for α ∈ (0, Proof. The proof of this theorem is based on the same Hölderian framework as the proof of Theorem 3 using Theorem 3 in [16] instead of Theorem 1 in [16] and Lemma A.1.

Consistency of test statistics
We investigate the consistency of the test statistics T α,n . The practical results are given in Corollaries 2 and 1. Proofs of these corollaries are based on the following generic result (Theorem 5) which has a broader scope. The consistency condition is expressed in terms of T α,n (τ n,1 , . . . , τ n,n ) = max where the τ n,k 's are defined by (3).
It is enough to write the proof for the case a n = 1, since in the general case, all the computations made below remain valid with a n in factor.
Let us compute n k=1 τ n,k .
Remark 1. From a statistical point of view, it is useful to find for which values of the parameter γ condition (31) does not induce some extra restriction on the choice of the sequence (m * (n)) n 1 . Writing θ n := m * (n)/n, we see that (31) is not satisfied if and only if there exists some subsequence (θ nj ) j 1 in (0, 1) such that e γ(1−θn j ) tends to 1 + γ/2. Then any θ limit of some subsequence of (θ nj ) j 1 (there is at least one such θ by compactness of [0, 1]) must satisfy 1 + γ/2 = e γ(1−θ) . Clearly, this equation has no solution for γ −2. For −2 < γ < 0, it has a unique solution It is easily seen that this solution belongs to [0, 1] only if γ 0 γ < 0, where γ 0 −1.5937. From this we can conclude that if γ < γ 0 , the condition (31) is satisfied without any extra restrictions on the choice of the sequence (m * (n)) n 1 . For γ 0 γ < 0, one can always find a sequence (m * (n)) n 1 for which (31) fails.

Remark 2.
From the consistency condition, one can see that the bigger α the shorter change can be detected with the statistics. As expected, the detection is not so good as in the i.i.d. case, see [12].
Proof. We keep the notations A and B already used in the previous proof. By Theorem 3, under (H 0 ), b n T α,n converges in distribution and hence is stochastically bounded for the normalization b n = n −3/2+α . So it remains only to check condition (25). This requires an estimate for the asymptotic order of magnitude of Using the second order expansions So the divergence (25) follows from the condition n −3/2+α * 2−α a n → ∞ as n → ∞ and (31).
Then under (H A ), The conclusion extends to the special case α = 0, under the same assumptions provided that (5), is replaced by E 2 1 < ∞.
Proof. By Theorem 4, under (H 0 ), b n T α,n converges in distribution and hence is stochastically bounded for the normalization b n = n −1/2+α (1−φ n ). So it remains only to check condition (25) in the three considered cases.
and recalling that lim sup * /n < 1, we immediately see that, for n large enough, there is some positive constant c such that Then the divergence (25) follows clearly from the condition n −1/2+α * 1−α a n → ∞ as n → ∞.
Remark 3. The graphical interpretation presented in Fig. 1 may provide a better understanding of the results in Corollary 2. Assume for simplicity that a n = 1, * n a (that is there are positive constants c 1 and c 2 such that, for n large enough, c 1 n a * c 2 n a ) and that φ n n b for some 0 < a, b < 1. For a given value of p in condition (5), what are the pairs (a, b) for which Corollary 2 allows detection of an epidemics of length * n a , subject to an admissible choice of α? The set of solutions is represented by the shadowed area of the unit square. The light grey part above the diagonal corresponds to Cases 1 and 2, that is lim n→∞ * (1 − φ n ) belongs to (0, ∞]. The triangular darker grey area corresponds to the case where * (1 − φ n ) tends to 0. One can remark that, when p tends to infinity, the whole shadowed area converges to the trapezoid with upper basis the upper side of the unit square and lower basis the segment [2/3, 1] on the horizontal axis.

Test power analysis
Here we perform the test power analysis. For this, we present the results of experiments in the Tables 1 and 2. We computed empirical power on size-adjusted (not nominal size) basis, i.e., replaced the nominal value of significance level by the value of empirical distribution function for p-values under null hypothesis. For more details on size power curves, see [17].
For different values of parameters γ, γ n , α, k * , * and a n , we compute N = 1000 realizations of test statistics with the sample size n. Innovations have been generated as standard normally distributed random variables. For the limit distribution, we compute N = 5000 realizations of test statistics with the sample size n = 5000. We approximate the values of the standard Wiener process by where (j) are generated as standard normally distributed random variables. The Ornstein-Uhlenbeck process have been approximated by the the following discretization: and S 0 = 0. For more details about (33), see [18]. Using values generated by (33), we approximate the integrated Ornstein-Uhlenbeck process by J k 5000 = n −1 k j=1 S j , k = 1, . . . , 5000.
Further modifying the separate parameters we compute the empirical size-power. We always keep all these parameters except one (indicated in the first column in both tables) which we allow to vary. Note that, in order to compute the test power, we need to compute the empirical p-values. Usually, the estimate of empirical p-value is p = s/N , where s is the number of values (limit process) that are greater than or equal to the observed value (statistics), N is the number of values. Nevertheless, the previous formula is biased due to the finite sampling. Davison and Hinkley [19,p. 141] suggested to correct the bias with such formula p = (s + 1)/(N + 1). One can observe, that these two formulas are essentially the same when the number of replications N is large, but we use unbiased estimate in this computations. As one can see in Table 1 the test power is almost the same for all α. The test power increases with the length of epidemics, but it has no big difference with increasing α. Note that, for the first type model, the location of epidemics makes the difference. The biggest power is for the epidemics in the middle of the observations. For this model, the test can detect the epidemic change best when a n = 1 or bigger, for the smaller changes, it has a lower power. Naturally, the test power increases with the number of observations and α. Further the bigger is γ, the bigger is test power. That is the test power increases when the ratio γ/n increases.
For the second type model (Table 2), the test power for all parameter values is the lowest, when α = 0 and increases with the α. For this model, detection of epidemic changes becomes better with the increasing length of epidemics, but the test detects short epidemic change very good for the bigger α. Note that the test power does not depend on the place of epidemics. Also, it detects quite good even small changes as a n = 0.8. The test power increases when the number of observations and α are increasing. The test power does not vary to much depending on γ n .
Let us introduce the closed subspace G α = {x ∈ H o α [0, 1]: x(0) = 0}. From (A.4) we see that the functional q = I α (·, s, t) satisfies on G α the Condition (iii) of Lemma A.3 with the constant C = 2. It follows by Lemma A.3 that g n as well as g are Lipschitz on G α with the same constant C. In fact, the sequence (g n ) n 2 is equicontinuous on G α .
Suppose that the sequence (η n ) n 1 is tight on C[0, 1] or on H o α [0, 1]. As the pointwise convergence of (g n ) is established, using Lemma A.2 we obtain that g n (η n ) = g(η n ) + o P (1).
The two following lemmas one can find in [12].
Lemma A.2. Let (η n ) be a tight sequence of random elements in separable Banach space B and g n , g be continuous functionals B → R. Assume that g n converges pointwise to g on B and that (g n ) is equicontinuous. Then g n (η n ) = g(η n ) + o P (1). Lemma A.3. Let (B, · ) be a vector normed space and q : B → R such that: (i) q is subadditive: q(x + y) q(x) + q(y), x, y ∈ B; (ii) q is symmetric: q(−x) = q(x), x ∈ B; (iii) for some constant C, q(x) C x , x ∈ B.
Then q satisfies the Lipschitz condition q(x + y) − q(x) C y , x, y ∈ B. (A.5) If F is any set of functionals q fulfilling (i), (ii) and (iii) with the same constant C, then (i), (ii) and (iii) are satisfied by g(x) := sup{q(x): q ∈ F} which therefore satisfies (A.5).