I Introduction
Quickest change detection (QCD) is a fundamental problem in mathematical statistics (see, e.g., [27] for an overview). Given a stochastic sequence whose distribution changes at some unknown changepoint, the goal is to detect the change after it occurs as quickly as possible, subject to false alarm constraints. The QCD framework has seen a wide range of applications, including lineoutage in power systems [2], dimtarget manoeuvre detection [13], stochastic process control [23], structural health monitoring [3], and piecewise stationary multiarmed bandits [1]. The two main formulations of the classical QCD problem are the Bayesian formulation [20, 25], where the changepoint is assumed to follow a known prior distribution, and the minimax formulation [10, 18], where the worstcase detection delay is minimized over all possible changepoints, subject to false alarm constraints. In both the Bayesian and minimax settings, if the pre and postchange distributions are known, lowcomplexity efficient solutions to the QCD problem can be found [27].
In many practical situations, we may not know the exact distribution in the pre or postchange regimes. While it is reasonable to assume that we can obtain a large amount of data in the prechange regime, this may not be the case for the postchange regime. Also, in applications such epidemic monitoring and piecewise stationary multiarmed bandits, a change in a specific statistic (e.g., the mean) of the distribution is of interest. This is different from the original QCD problem where any distributional change needs to be detected. Furthermore, in many applications, the support of the distribution is bounded. For example, the observations representing the fraction of some specific group in the entire population are bounded between 0 and 1. This is the case, for example, in the pandemic monitoring problem that we discuss in detail in Section IV. In many applications, including the pandemic monitoring problem, the system has usually reached some nominal steadystate distribution before the changepoint. In these situations, the prechange distribution can be assumed to be stationary.
In this paper, we study the problem of quickest detection of a change in the mean of a sequence of independent observations. The prechange distribution is assumed to be stationary, while the postchange distributions are allowed to be nonstationary. We first study the case where the prechange distribution is known, and then study the extension where only the mean and variance of the prechange distribution are known. No knowledge of the postchange distributions is assumed other than that their means are above some threshold larger than the prechange mean.
There have been a number of lines of work on the QCD problem when the pre and/or postchange distributions are not completely known. The most prevalent is the generalized likelihood ratio (GLR) approach, introduced in [10]
for the parametric case where the postchange distribution has an unknown parameter. This GLR approach is studied in detail for the problem of detecting the change in the mean of a Gaussian distribution with unknown postchange mean in
[21]. A GLR test for the case where the pre and postchange distributions come from an oneparameter exponential family, and both the pre and postchange parameters are unknown, is analyzed in [7].The QCD problem has also been studied in a nonparametric setting. In particular, for detecting a change in the mean of an observation sequence, one approach has been to use maximum scan statistics. The scan statistic of an observation sequence is defined as the absolute difference of the averages before and after a potential changepoint. In [4]
, the case where the pre and postchange distributions have finite moment generating functions in some neighborhood around zero is considered. At each time greater than a window size
, the scan statistic at each potential changepoint is calculated using the last observations. The maximum scan statistic is then calculated over the set of potential changepoints, and an alarm is raised if this maximum exceeds some threshold. In [12], the case of subGaussian pre and postchange distributions is studied. The scan statistic is calculated over the entire observation sequence, and the maximum is compared to a threshold determined by the current time and the desired false alarm rate. This approach is further applied to the piecewise stationary multiarmed bandit problem in [1]. We compare our approach to meanchange detection with a test using scan statistics in Section IV.We note that for both the GLR the scan statistics approaches, the complexity of computing the test statistic at each timestep grows at least linearly with the number of samples. In practice, a windowed version of the test statistic is often used to reduce computational complexity, while suffering some loss in performance.
Still another line of work is the one based on a minimax robust approach [5], in which it is assumed that the distributions come from mutually exclusive uncertainty classes. Under certain conditions on the uncertainty classes, e.g., joint stochastic boundedness [15], lowcomplexity solutions to the minimax robust QCD problem can be found [26]. Under more general conditions, e.g., weak stochastic boundedness, a solution that is asymptotically close to the minimax solution can be found [13].
In this paper, we use an asymptotic version of the minimax robust QCD problem formulation [13] to develop algorithms for the nonparametric detection of a change in mean of an observation sequence. Our contributions are as follows:

We extend the asymptotic minimax robust QCD problem introduced in [13] to the more general nonstationary setting.

We study the problem of quickest detection of a change in the mean of an observation sequence under the assumption that no knowledge of the postchange distribution is available other than that its mean is above some threshold larger than the prechange mean.

For the case where the prechange distribution is known, we derive a test that asymptotically minimizes the worstcase detection delay over all possible postchange distributions, as the false alarm rate goes to zero.

We study the limiting form of the optimal test as the gap between the pre and postchange means goes to zero, which we call the MeanChange Test (MCT). We show that the MCT can be designed with only knowledge of the mean and variance of the prechange distribution.

We also characterize the performance of the MCT when the mean gap is moderate, under the assumption that the distributions of the observations have bounded support.

We validate our analysis through numerical results for detecting a change in the mean of a beta distribution. We also demonstrate the use of the MCT for pandemic monitoring.
The rest of the paper is structured as follows. In Section II, we describe the quickest change detection problem under distributional uncertainty and provide some new results regarding asymptotically robust tests in the nonstationary setting. In Section III, we formulate the mean change detection problem, and propose and analyze the meanchange test (MCT), which solves the problem asymptotically. In Section IV, we validate our analysis through numerical results for detecting a change in the mean of a beta distribution, and also demonstrate the use of the MCT in monitoring pandemics. Finally, in Section V, we provide some concluding remarks.
Ii Quickest Change Detection Under Distributional Uncertainty
Let
be a sequence of independent random variables, and let
be a changepoint. Let andbe two sequences of probability measures, where
and for all . Further, assume that has probability density with respect to the Lebesgue measure on , for and . Let denote the probability measure on the entire sequence of observations when the prechange distributions are and the postchange distributions are , with and , and let denote the corresponding expectation. When and are stationary, i.e., , and , , we use the notations and in place of and , respectively.The changetime is assumed to be unknown but deterministic. The problem is to detect the change quickly while not causing too many false alarms. Let be a stopping time [15] defined on the observation sequence associated with the detection rule, i.e. is the time at which we stop taking observations and declare that the change has occurred.
For the case where both the pre and postchange distributions are stationary and known, Lorden [10] proposed solving the following optimization problem to find the best stopping time :
(1) 
where
(2) 
is a worstcase delay metric, and
(3) 
with
(4) 
Here is the expectation operator when the change never happens, and .
Lorden also showed that Page’s Cumulative Sum (CuSum) algorithm [17] whose test statistic is given by:
(5) 
solves the problem in (1) asymptotically. Here is the likelihood ratio:
(6) 
The CuSum stopping rule is given by:
(7) 
where . It was shown by Moustakides [16] that the CuSum algorithm is exactly optimal for the problem in (1).
When the prechange and postchange distributions are unknown but belong to known uncertainty sets and are possibly nonstationary, a minimax robust formulation can be used in place of (1):
(8) 
where
(9) 
and the feasible set is defined as
(10) 
with
(11) 
We now address the solution to the problem in (8). To this end, we give the following using definitions.
Definition II.1.
If the pair of pre and postchange uncertainty sets is JS bounded, the CuSum test statistic (see (II)), with stopping rule (see (7)), solves (8) exactly both when and are stationary [26] and when they are potentially nonstationary [14].
Definition II.2.
(see [13]) A pair of uncertainty sets is said to be weakly stochastically (WS) bounded by if
(14) 
for all , and
(15) 
for all . Here, denotes the expectation operator with respect to distribution , and denotes KLdivergence:
(16) 
It is shown in [13] that if the pair of uncertainty sets is JS bounded by , it is also WS bounded by . It is also shown in [13] that if the pair of pre and postchange uncertainty sets is WS bounded, the CuSum test statistic with stopping rule solves (8) asymptotically as when and are both stationary.
Iia Asymptotically Optimal Solution in the Nonstationary Setting
Let be such that is WS bounded by . In the following, we extend the result in [13] to the case where and are potentially nonstationary and derive an asymptotically optimal solution as . Specifically, through Lemma II.1 we upper bound the asymptotic delay, through Lemma II.2 we control the false alarm rate, and in Theorem II.3 we combine the lemmas to provide an asymptotically optimal solution to the problem in (8) when and are potentially nonstationary.
Lemma II.1.
Consider WS bounded by . Let and be such that and for all . Suppose that for all ,
where denotes the variance of when . Then, satisfies
(17) 
as , where as .
Lemma II.2.
Theorem II.3.
Iii MeanChange Detection Problem
Until now, we have considered the general QCD problem formulated in (8). In this paper, we are mainly interested in a special case of the problem, described as follows. The prechange distribution is stationary, i.e., , with prechange mean and variance . Thus, is a singleton. The postchange distribution could be nonstationary, and at each time it belongs to the following uncertainty set:
(20) 
In this expression, denotes a generic observation in the sequence, and is a predesigned threshold. Define
(21) 
which is half of the worstcase meanchange gap.
The minimax robust meanchange problem, which is a reformulation of (8) is given by:
(22) 
Our goal is to find a stopping time that solves (22) asymptotically as the false alarm rate .
Iiia Known Prechange Distribution
Define
(23) 
to be the cumulantgenerating function (cgf) of the observations under . In the following theorem, we provide a solution to the problem stated in (22).
Theorem III.1.
Proof.
The proof follows from an application of Theorem II.3 if we can establish that is WS bounded by . By [13, Prop. 1 (iii)], since is convex and is a singleton, if minimizes the KLdivergence over , then is WS bounded by . Therefore, it remains to show that specified in (24) minimizes , subject to . To this end, we follow the procedure outlined in [8, Sec. 6.4.1]. Consider the Lagrangian
(28) 
where the Lagrange multiplier corresponds to the constraint that the postchange mean is greater than , and corresponds to the constraint that is a probability measure. For an arbitrary direction , we take the Gateaux derivative with respect to :
(29) 
where , and since is arbitrary, we arrive at
(30) 
By the Generalized Kuhn–Tucker Theorem [11], since is bounded, is a necessary condition for optimality. Furthermore, since is convex in , this is also a global optimum. To satisfy the constraints, we have
(31) 
and that satisfies
(32) 
Thus, in (24) minimizes , subject to .
Furthermore, the minimum KLdivergence is
(33) 
Hence, the worstcase delay satisfies
(34) 
as . ∎
Note that is an exponentiallytilted version (or the Esscher transform) of .
IiiB Approximation for Small
Even though we have an expression for the test statistic when is known, as given in (26), the exact solution of is not available in closedform. Fortunately, if the meanchange gap is small, we obtain a lowcomplexity test in terms of only the prechange mean and variance that closely approximates the performance of the asymptotically minimax optimal test in the previous section.
As , , and hence . From a secondorder Taylor expansion on around 0, we obtain
(35) 
In this same regime, by continuity of ,
(36) 
where we have used . Hence, the approximate loglikelihood ratio at time is
(37) 
and the corresponding minimum KLdivergence is approximated as:
(38) 
Now
(39) 
where
(40) 
Therefore, the stopping rule can be approximated by the stopping rule , where
(41) 
with . We call the MeanChange Test (MCT), and the MCT statistic.
From (38), it follows that as and , the worstcase delay satisfies
(42) 
Therefore, if is small, it is sufficient to know only the mean and variance to construct a good approximation to the asymptotically minimax robust test. Furthermore, only the mean of the prechange distribution is needed to construct the MCT statistic. From the simulation results in Section IV, we see that the performance of the MCT can be very close to that of the asymptotically minimax robust test even for moderate values of
. Since the mean and variance of a distribution are much easier and more accurate to estimate than the entire density, this test can be useful and accurate when only a moderate number of observations in the prechange regime is available.
IiiC Performance Analysis of MCT for moderate
We now study the asymptotic performance of the MCT for fixed , as . For this part of the analysis, we assume that the pre and postchange distributions have supports that are uniformly bounded, and without loss of generality, we assume that the bounding interval is . This assumption holds in many practical applications, including the pandemic monitoring problem discussed in Section IV.
Define
(43) 
Then the MCT statistic of (IIIB) can be written as:
(44) 
with . The MCT stopping time is given by:
(45) 
where has to be chosen to meet the FAR constraint:
(46) 
In what follows, we write as , with the understanding that the test statistic being used throughout is the MCT statistic .
IiiC1 False Alarm Analysis
In Lemma III.2 below, we first control the boundary crossing probability of in the prechange regime. Then, in Theorem III.3, we use Lemma III.2 to bound the false alarm rate of the MCT asymptotically using the procedure outlined in [22].
Lemma III.2.
Assume that the prechange distribution has known prechange mean and variance , and that the postchange distribution is nonstationary with , for all . For , define the supplementary stopping time
(47) 
where , with defined in (43). Then,
(48) 
where
(49) 
and is the modified Bessel function of the second kind of order .
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