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We consider the quickest change-point detection problem where the aim is to detect the onset of a prespecified drift in “live”-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The object of interest is the distribution of the stopping time associated with the Generalized Shryaev–Roberts (GSR) detection procedure set up to “sense” the presence of the drift in the Brownian motion under surveillance. Specifically, we seek the GSR stopping time's survival function (the tail probability that no alarm is triggered by the GSR procedure prior to a given point in time), and distinguish two scenarios: (a) when the drift never sets in (prechange regime) and (b) when the drift is in effect ab initio (postchange regime). Under each scenario, we obtain a closed-form formula for the respective survival function, with the GSR statistic's (deterministic) nonnegative headstart assumed arbitrarily given. The two formulae are found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. We then exploit the obtained formulae numerically and characterize the pre- and postchange distributions of the GSR stopping time depending on three factors: (1) magnitude of the drift, (2) detection threshold, and (3) the GSR statistic's headstart. [ABSTRACT FROM PUBLISHER]

In sequential analysis, an experimenter gathers information regarding an unknown functional (parameter) by observing random samples in successive steps. We discuss a number of distribution-free scenarios under a variety of loss functions. The number of observations gathered upon termination is a positive integer-valued random variable, customarily denoted by N. Often, a standardized version of N would follow an approximate normal distribution in the asymptotic sense. We provide exploratory data analysis (EDA) with the help of a number of interesting illustrations. We do so via large-scale simulation studies to demonstrate broad applicability of the purely sequential methodologies along with the appropriateness of asymptotic normality of the standardized stopping variables as a practical and useful guideline. [ABSTRACT FROM AUTHOR]

In early phase cancer clinical trials where toxicity events follow independent and identical Bernoulli distributions indexed by patients, the Bayesian stopping rule has been used for continuous monitoring of toxicity along with an affordable maximum sample size (N). This article studies some properties of an heuristic procedure where the trial will stop at the first time that the posterior probability that the toxicity rate (p) is greater than a threshold (η) is greater than certain probability threshold (τ). Specifically, we study the pattern formed by stopping times and regions, recursive stopping probability computation, and toxicity rate estimation. Some relevant theoretical results are given. The presented results are potentially useful for guiding toxicity clinical trial designs. [ABSTRACT FROM AUTHOR]

Consider a zero-mean and second-order stationary time series of interest that cannot be observed directly. Instead an amplitude-modulated time series is observed where is a stationary Bernoulli time series and is a time series of independent variables satisfying and Time series creates missing observations when At = 0, and Ut modulates not missed Xt. There is bad and good news about spectral analysis of amplitude-modulated time series. The bad news is that in general consistent estimation of the spectral density is impossible. The good news is that the spectral shape (which is the spectral density minus) multiplied by factor may be consistently estimated. This article, for the first time in the literature, explores a classical problem of sequential nonparametric estimation of the scaled shape with assigned mean integrated square error. It proposes an adaptive sequential estimator that solves the problem and whose mean stopping time matches the performance of a minimax oracle that knows an underlying spectral density and the amplitude-modulating mechanism. The asymptotic theory is complemented by numerical examples. [ABSTRACT FROM AUTHOR]

Motivated by the practical investigation of a state-dependent quickest detection problem in continuous time, especially for Brownian observations, we propose an asymptotic scheme in discrete time called a quickest detection scheme of an accumulated state-dependent change point. Here the state-dependent means that the priori probability of the change point depends on the current state. We reduce the problem to finding an optimal stopping time of a vector-valued Markov process. We illustrate the scheme via a numerical example. [ABSTRACT FROM AUTHOR]

A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formulated weighted squared error loss (SEL) due to estimation of g (μ) , a function of μ, with g ( X ¯ N) plus linear cost of sampling from a N (μ , σ 2) population having both parameters unknown. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, → 0 ) results is laid out including the first-order and second-order efficiency properties. Then, accurate sequential risk calculations are launched, which are then followed by two main results: (i) Theorem 4.1 shows an asymptotic risk efficiency property, and (ii) Theorem 5.1 shows an asymptotic second-order regret expansion associated with the proposed purely sequential MRPE strategy assuming suitable conditions on g(.). We also provide a bias-corrected version of the terminal estimator, g ( X ¯ N). We follow up with a number of interesting illustrations where Theorems 4.1–5.1 are readily exploited to conclude an asymptotic risk efficiency property and second-order regret expansion, respectively. A number of other interesting illustrations are highlighted where it is possible to verify the conclusions from Theorems 4.1–5.1 more directly with less stringent assumptions on the pilot sample size. [ABSTRACT FROM AUTHOR]

Sampling strategies with fractional acceleration 0 < ρ < 1 achieve substantial operational savings compared with purely sequential counterparts. But, acceleration customarily yielded second-order (s.o.) lower and upper bounds for requisite characteristics when ρ − 1 was not an integer. First time in the literature, we have recently designed acceleration with asymptotic s.o. expansions in normal mean problems with ρ − 1 arbitrary. In this article, a general unified theory is now developed leading to asymptotic s.o. expansions for customarily studied characteristics with arbitrary ρ − 1 . That is, the previously known s.o. lower/upper bounds can be replaced with appropriate s.o. expansions in a variety of inference problems with prescribed accuracy. We emphasize the theoretical foundation and ensuing analyses with a series of interesting inference problems from (a) a number of non-normal distributions as well as (b) one- and two-sample distribution-free scenarios. These prove a desirable breadth of coverage under our proposed big tent. [ABSTRACT FROM AUTHOR]

The problem of multihypothesis testing with controlled sensing of observations is considered. The distribution of observations collected under each control is assumed to follow a single-parameter exponential family distribution. The goal is to design a policy to find the true hypothesis with minimum expected delay while ensuring that the probability of error is below a given constraint. The decision-maker can reduce the delay by intelligently choosing the control for observation collection in each time slot. A policy for this problem is derived that satisfies given constraints on the error probability, and it is shown that this policy is asymptotically optimal in the sense that it asymptotically achieves an information-theoretic lower bound on the expected delay. Numerical results are provided that illustrate an application of the policy to medical diagnostic inference. [ABSTRACT FROM AUTHOR]

The problem of asymptotic efficiency of adaptive one-step predictors for a stable multivariate first-order autoregressive process (AR(1)) with unknown parameters is considered. The predictors are based on the truncated estimators of the dynamic matrix parameter. The truncated estimation method is a modification of the truncated sequential estimation method that makes it possible to obtain estimators of ratio-type functionals with a given accuracy by samples of fixed size. The criterion of optimality is based on the loss function, defined as a sum of sample size and squared prediction error's sample mean. The cases of known and unknown variance of the noise model are studied. In the latter case the optimal sample size is a special stopping time. The simulation results are given. [ABSTRACT FROM AUTHOR]

A vector-valued autoregressive time series model is considered. The autoregressive coefficients of the model are random with possible dependencies among them. Estimation of the large number of parameters in such models becomes costly with an increase in dimension. A sequential procedure is proposed that promises a significant gain in the sample size thus reduction in the cost of implementation. The procedure is also risk efficient in the sense that as the cost of sampling becomes negligible the asymptotic predictive risk of the proposed procedure reaches the oracle predictive risk corresponding to the best fixed sample size procedure that assumes the values of the nuisance parameters to be known. Extensive simulation results are presented to illustrate the properties of the proposed procedure in a finite sample. [ABSTRACT FROM AUTHOR]

We study the problem of detecting a drift change of a Brownian motion under various extensions of the classical case. Specifically, we consider the case of a random post-change drift and examine monotonicity properties of the solution with respect to different model parameters. Moreover, robustness properties – effects of misspecification of the underlying model – are explored. [ABSTRACT FROM AUTHOR]

In this article, we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimizing the expected sample size (ESS) under restrictions on the error probabilities. We take, as a criterion of minimization, a weighted sum of the ESSs evaluated at some points of interest in the parameter space, aiming at its minimization under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers based on the minimization of an auxiliary objective function (called Lagrangian) combining the objective function with the restrictions, taken with some constants called multipliers. Subsequently, the multipliers are used to make the solution comply with the restrictions. We develop a computer-oriented method of minimization of the Lagrangian function that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, as in Bayesian, Kiefer-Weiss, and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population, we develop a set of computer algorithms for designing sequential tests that minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS. We implement the algorithms in the R programming language. The program code is available in a public GitHub repository. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out. A method of solution of the multi-hypothesis Kiefer-Weiss problem is proposed and is applied for a particular case of three hypotheses in the Bernoulli model. [ABSTRACT FROM AUTHOR]

In this article, we deal with problems of testing hypotheses in the framework of sequential statistical analysis. The main concern is the optimal design and performance evaluation of sampling plans in Kiefer-Weiss problems. The main goal of the Kiefer-Weiss problem is designing hypothesis tests that minimize the maximum average sample number, over all parameter values, as opposed to both the sequential probability tests (SPRTs) minimizing the average sample number only at two hypothesis points and the classical fixed-sample-size test. For observations that follow a distribution from an exponential family of the continuous type, we provide algorithms for optimal design in the modified Kiefer-Weiss problem and obtain formulas for evaluating the performance of sequential tests by calculating the operating characteristic function, the average sample number, and some related characteristics. These formulas cover, as a particular case, the SPRTs and their truncated versions, as well as optimal finite-horizon sequential tests. In the setting of the original Kiefer-Weiss problem we apply the method of our recent work (Sequential Analysis 2022, 41(2), 198–219) for numerical construction of the optimal tests. For the particular case of sampling from a normal distribution with a known variance, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test provided that the three tests have the same levels of the error probabilities. All of the algorithms are implemented in the form of computer code written in the R programming language and are available at the GitHub public repository (). Guidelines on the adaptation of the program code to other exponential family distributions are provided. [ABSTRACT FROM AUTHOR]

This article addresses the sequential detection of transient changes by using the finite moving average (FMA) test. It is assumed that a change occurs at an unknown (but nonrandom) change point and the duration of postchange period is finite and known. We relax the assumption that the profile of a transient change is chosen so that the log-likelihood ratios of the observations are associated random variables (r.v.s) in the prechange mode. Hence, the profile of the transient change is arbitrary and it is not necessarily of constant sign for a distribution with monotone likelihood ratio. A new upper bound for the worst-case probability of false alarm is proposed. It is shown that the optimization of the window-limited cumulative sum (CUSUM) test again leads to the FMA test. Three particular transient changes are considered: in the Gaussian mean, in the Gaussian variance, and in the parameter of exponential distribution. In the first case, a comparison between the bounds for the FMA test operating characteristics and the exact operating characteristics calculated by numerical integration is used to estimate the sharpness of the bounds. In the second and third cases, special attention is paid to the calculation of the FMA distribution in the case of arbitrary profile. The method of convolution is used to solve the problem. [ABSTRACT FROM AUTHOR]

In this article, we deal with problems of testing hypotheses in the framework of sequential statistical analysis. The main concern is the optimal design and performance evaluation of sampling plans in the Kiefer-Weiss problems. For the case of observations following a discrete exponential family, we provide algorithms for optimal design in the modified Kiefer-Weiss problem and obtain formulas for evaluating their performance, calculating operating characteristic function, average sample number, and some related characteristics. These formulas cover, as a particular case, sequential probability ratio tests (SPRTs) and their truncated versions, as well as optimal finite-horizon sequential tests. On the basis of the developed algorithms we propose a method of construction of optimal tests and their performance evaluation for the original Kiefer-Weiss problem. All algorithms are implemented as functions in the R programming language and can be downloaded from , where the code for the binomial, Poisson, and negative binomial distributions is readily available. Finally, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test having the same levels of the error probabilities. [ABSTRACT FROM AUTHOR]

We develop a new class of purely sequential methodologies under an assumption that the population distribution belongs to a location-scale family. Both asymptotic first-order and second-order theories are put forward with substantial generality under a big and unified tent that successfully lead to a broad set of illustrations. After we identify an appropriately defined optimal strategy under this unified structure, we introduce applications that handle a variety of interesting inference problems. These are associated with, but not limited to, the following areas: (a) the fixed-width confidence interval (FWCI) estimation, (b) the minimum risk point estimation (MRPE), (c) the fixed-size confidence region (FSCR) estimation, (d) multiple comparisons, and (e) selecting the best normal treatment (StBNT). In illustrations (a)–(d), we have highlighted a number of choices of population distributions. Some illustrations are accompanied with data analyses. [ABSTRACT FROM AUTHOR]

In this article, we propose a modified max-cumulative sum (CUSUM) procedure for detecting changes in parameters of skew normal distribution. The corresponding false alarms frequency and the postchange detection delay are investigated. Asymptotic behaviors of detection delay and theoretical optimality of the detection procedure have been established. Simulations have been conducted to show the performance of the proposed method and compare it to the other existing methods including CUSUM. Real data are given to illustrate the detection procedure. [ABSTRACT FROM AUTHOR]

In this discussion, we focus on a sequential procedure for point and confidence region estimating parameters in autoregressive processes AR(1). The uniform asymptotic normality of estimators is considered. [ABSTRACT FROM AUTHOR]

Change-of-measure is a powerful technique in wide use across statistics, probability, and analysis. Particularly known as Wald's likelihood ratio identity, the technique enabled the proof of a number of exact and asymptotic optimality results pertaining to the problem of quickest change-point detection. Within the latter problem's context we apply the technique to develop a numerical method to compute the generalized Shiryaev–Roberts (GSR) detection procedure's pre-change run length distribution. Specifically, the method is based on the integral equations approach and uses the collocation framework with the basis functions chosen to exploit a certain change-of-measure identity and a specific martingale property of the GSR procedure's detection statistic. As a result, the method's accuracy and robustness improve substantially, even though the method's theoretical rate of convergence is shown to be merely quadratic. A tight upper bound on the method's error is supplied as well. The method is not restricted to a particular data distribution or to a specific value of the GSR detection statistic's head start. To conclude, we offer a case study to demonstrate the proposed method at work, drawing particular attention to the method's accuracy and its robustness with respect to three factors: (1) partition size (rough vs. fine), (2) change magnitude (faint vs. contrast), and (3) average run length (ARL) to false alarm level (low vs. high). Specifically, assuming independent standard Gaussian observations undergoing a surge in the mean, we employ the method to study the GSR procedure's run length's pre-change distribution, its average (i.e., the usual ARL to false alarm), and its standard deviation. As expected from the theoretical analysis, the method's high accuracy and robustness with respect to the foregoing three factors are confirmed experimentally. We also comment on extending the method to handle other performance measures and other procedures. [ABSTRACT FROM PUBLISHER]

We consider the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution. This problem is investigated in a sequential setup under mild assumptions on the underlying random process. The optimal method minimizes the expected number of samples while ensuring that the average detection/estimation errors do not exceed a certain level. After converting the constrained problem to an unconstrained one, we characterize the general solution by a nonlinear Bellman equation, which is parameterized by a set of cost coefficients. A strong connection between the derivatives of the cost function with respect to the coefficients and the detection/estimation errors of the sequential procedure is derived. Based on this fundamental property, we further show that for suitably chosen cost coefficients the solutions of the constrained and the unconstrained problem coincide. We present two approaches to finding the optimal coefficients. For the first approach, the final optimization problem is converted into a linear program, whereas the second approach solves it with a projected gradient ascent. To illustrate the theoretical results, we consider two problems for which the optimal schemes are designed numerically. Using Monte Carlo simulations, it is validated that the numerical results agree with the theory. [ABSTRACT FROM AUTHOR]

Research on classical fixed-width confidence interval (FWCI) estimation problems for a normal mean when the variance remains unknown have steadily moved along under a zero-one loss function. On the other hand, minimum risk point estimation (MRPE) problems have grown largely under a squared error loss function plus sampling cost. However, the FWCI problems customarily do not take into account any sampling cost in their formulations. This fundamental difference between the two treatments has led the literature on the FWCI and MRPE problems to grow in multiple directions in their own separate ways from one another. In this article, we introduce a new formulation combining both MRPE and FWCI methodologies with desired asymptotic first-order (Theorems 2.1–2.2) and asymptotic second-order characteristics (Theorem 2.3) under a single unified structure, allowing us to develop a genuine minimum risk fixed-width confidence interval (MRFWCI) estimation strategy. Fruitful ideas are proposed by incorporating illustrations from purely sequential and other multistage MRFWCI problems with an explicit presence of a cost function incurred due to sampling. We supplement the general theory and methodology by means of illustrations and analyses from simulated data along with applications to air quality data. [ABSTRACT FROM AUTHOR]

In this article, motivated by detecting the occurrence of an epidemic when the arrival rates of patients increase in a portion of M panels or detecting the deterioration of a system composed of M independent components that causes an increase in failure rates in a portion of components, we consider the detection of a common change when M independent Poisson processes are monitored simultaneously where only a portion of the processes have rate increases after the change time. M individual cumulative sum (CUSUM) processes and Shiryaev-Roberts (S-R) processes are calculated recursively in parallel at each pooled arrival time. A systematic procedure is proposed by using the sum of M S-R processes as the detection process for a common change. After the detection, the M individual CUSUM processes are used to isolate the changed panels with false discovery rate (FDR) control and then the medians of the change time estimates from each individual CUSUM process or S-R process based on the isolated panels are used to estimate the common change time. The model can be generalized to different prechange rates, jittered change time, and unknown postchange rates. [ABSTRACT FROM AUTHOR]

We consider sequential hypothesis testing based on observations that are received in groups of random size. The observations are assumed to be independent both within and between the groups. We assume that the group sizes are independent and their distributions are known and that the groups are formed independent of the observations. We are concerned with a problem of testing a simple hypothesis against a simple alternative. For any (group) sequential test, we take into account the following three characteristics: its type I and type II error probabilities and the average cost of observations. Under mild conditions, we characterize the structure of sequential tests minimizing the average cost of observations among all sequential tests whose type I and type II error probabilities do not exceed some prescribed levels. [ABSTRACT FROM AUTHOR]

We present a computational approach to the solution of the Kiefer-Weiss problem. Algorithms for construction of the optimal sampling plans and evaluation of their performance are proposed. In the particular case of Bernoulli observations, the proposed algorithms are implemented in the form of R program code. Using the developed computer program, we numerically compare the optimal tests with the respective sequential probability ratio test (SPRT) and the fixed sample size test for a wide range of hypothesized values and type I and type II errors. The results are compared with those of D. Freeman and L. Weiss (Journal of the American Statistical Association, 59, 1964). The R source code for the algorithms of construction of optimal sampling plans and evaluation of their characteristics is available at . [ABSTRACT FROM AUTHOR]

Most of the classical change-point detection schemes are designed for the sequences of independent and identically distributed (i.i.d.) random variables. In this article, motivated by the outbreak of 2009 H1N1 pandemic influenza, we develop change-point detection procedures for the susceptible–infected–recovered (SIR) epidemic model, where a change-point in the infection rate parameter signifies either the beginning or the end of an epidemic trend. The considered model falls into a general class ofbinomial thinningprocesses, which is a Markov chain. The cumulative sum (CUSUM) change-point detection procedure is developed for this class, and its performance is evaluated. Apparently, the CUSUM stopping rule is no longer optimal for this non-i.i.d. case. It can be improved by introducing a non constant adaptive threshold. The resulting modified scheme attains a shorter mean delay and at the same time a longer expected time of a false alarm, given that a false alarm eventually occurs. Proposed detection procedures are applied to the 2001–2012 influenza data published by the Centers for Disease Control and Prevention. [ABSTRACT FROM PUBLISHER]

The Shewhart control chart was first to monitor an ongoing process and raise an alarm when it appears that the level has changed. We show that the Shewhart chart is optimal for the criterion of maximizing the probability of detecting a change upon its occurrence subject to an average run length to false alarm. It is remarkable, particularly when the change is of moderate size, that Shewhart's procedure is better than cumulative sum (CUSUM). In the multivariate setting, applying the Shewhart procedure to each process separately is suboptimal. We create a generalized Shewhart procedure that is optimal for the aforementioned criterion. The results are illustrated in common settings. [ABSTRACT FROM PUBLISHER]

The problem of sequential hypothesis testing is studied, where samples are taken sequentially, and the goal is to distinguish between the null hypothesis where the samples are generated according to a distribution p and the alternative hypothesis where the samples are generated according to a distribution q. The defender (decision maker) aims to distinguish the two hypotheses using as few samples as possible subject to false alarm constraints. The problem is studied under the adversarial setting, where the data generating distributions under the two hypotheses are manipulated by an adversary, whose goal is to deteriorate the performance of the defender—for example, increasing the probability of error and expected sample sizes—with minimal cost. Specifically, under the null hypothesis, the adversary picks a distribution p ∈ P with cost c 0 (p) , and under the alternative hypothesis, the adversary picks a distribution q ∈ Q with cost c 1 (q). This problem is formulated as a non-zero-sum game between the defender and the adversary. A pair of strategies (the adversary's strategy and the defender's strategy) is proposed and proved to be a Nash equilibrium pair for the non-zero-sum game between the adversary and the defender asymptotically. The defender's strategy is a sequential probability ratio test and thus is computationally efficient for practical implementation. [ABSTRACT FROM AUTHOR]

We introduce and discuss some key inequalities that underlie the lower bound formula for the probability of lattice events in the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. The present work combines the notion of lattice events—as previously discussed for the nonadaptive member of the family—with the positive cumulative sum property for the adaptive members—as previously discussed for the special lattice event of correct selection, thereby extending the key inequality to its broadest scope. [ABSTRACT FROM AUTHOR]

We have designed multistage minimum risk point estimation (MRPE) strategies for a function of an unknown mean in a normal population with its variance unknown. These are developed under a weighted power absolute error loss (PAEL) function plus nonlinear cost of sampling by incorporating purely sequential, accelerated sequential, and three-stage estimation methodologies. Crucial asymptotic first-order and asymptotic second-order properties have been proved thoroughly under all three MRPE strategies. Extensive sets of simulations tend to validate nearly all desirable asymptotic properties for small to medium to large optimal fixed sample sizes. [ABSTRACT FROM AUTHOR]

This article develops a detection framework using Bayesian philosophy by adaptation of Shiryaev's and Roberts' methodology. We propose two unifying versions directly applicable in industrial process control and easily extendable to public health infectious disease surveillance via some data detrending and/or demodulation. The root idea uses the sum of likelihood ratios upon which an optimal stopping criterion is based. It sets a prior on the epoch of a change, allows the flexibility to elicit a prior distribution on other process parameters, and attempts to minimize an expected loss function. A sensitivity analysis is conducted for validation and performance assessment and analytical formulas are derived. The methods are successfully applied to the European Union Centre for Disease Control (ECDC) open-source global COVID-19 incidence data. We further lay out scenarios where interest may switch to the detection of separate outbreaks with similar syndromes during an already evolving epidemic. We view our approach as a toolkit with a potential to augment early reports to sentinels in syndromic surveillance and in biosurveillance. [ABSTRACT FROM AUTHOR]

In this article, we study sequential testing problems with overlapping hypotheses. We first focus on the simple problem of assessing if the mean μ of a Gaussian distribution is smaller or larger than a fixed ϵ > 0 ; if μ ∈ (− ϵ , ϵ) , both answers are considered to be correct. Then, we consider probably approximately correct best arm identification in a bandit model: given K probability distributions on R with means μ 1 , ... , μ K , we derive the asymptotic complexity of identifying, with risk at most δ, an index I ∈ { 1 , ... , K } such that μ I ≥ max i μ i − ϵ. We provide nonasymptotic bounds on the error of a parallel general likelihood ratio test, which can also be used for more general testing problems. We further propose a lower bound on the number of observations needed to identify a correct hypothesis. Those lower bounds rely on information-theoretic arguments, and specifically on two versions of a change of measure lemma (a high-level form and a low-level form) whose relative merits are discussed. [ABSTRACT FROM AUTHOR]

Purely sequential estimation for unknown mean ( μ) in a normal population having an unknown variance ( σ 2 ) when observations are gathered in groups has been recently discussed in Mukhopadhyay and Wang (2020). In this article, we briefly revisit two fundamental problems on sequential estimation: (i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem. However, we substitute the estimators defining the stopping boundaries with newly constructed unbiased and consistent estimators under permutations within each group. These new estimators incorporated in the definition of the stopping boundaries have led to tighter estimation of requisite optimal fixed sample sizes. We have analyzed the first-order and second-order asymptotic properties under appropriate requirements on the pilot size. Large-scale computer simulations and substantial data analysis have validated such first-order and second-order results. The methodologies are illustrated with the help of time series data on offshore wind energy. [ABSTRACT FROM AUTHOR]

We begin with a review of asymptotic properties of a purely sequential minimum risk point estimation (MRPE) methodology for an unknown mean in a one-parameter exponential distribution under a class of generalized loss functions. This class of powered absolute error loss (PAEL) includes both squared error loss (SEL) and absolute error loss (AEL) plus cost of sampling. We prove the asymptotic second-order efficiency property and asymptotic first-order risk efficiency property associated with the purely sequential MRPE problem. For operational convenience, we then move to implement an accelerated sequential MRPE methodology and prove the analogous asymptotic second-order efficiency property and asymptotic first-order risk efficiency property. We follow up with extensive data analysis from simulations and provide illustrations using cancer data. [ABSTRACT FROM AUTHOR]

For the classical Shiryaev-Roberts martingale diffusion considered on the interval [ 0 , A ] , where A > 0 is a given absorbing boundary, it is shown that the rate of convergence of the diffusion's quasi-stationary cumulative distribution function (c.d.f.), Q A (x) , to its stationary c.d.f., H(x), as A → + ∞ , is no worse than O ( log (A) / A) , uniformly in x ⩾ 0. The result is established explicitly by constructing new tight lower- and upper-bounds for Q A (x) using certain latest monotonicity properties of the modified Bessel K function involved in the exact closed-form formula for Q A (x) recently obtained by Polunchenko (2017b). [ABSTRACT FROM AUTHOR]

Two fundamental problems on purely sequential estimation are revisited—(i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem—in the context of estimating an unknown mean (μ) in a normal population having an unknown variance ( σ 2 ). We begin by laying down general frameworks for the second-order asymptotic analyses, in both problems, under sequential sampling of one observation at a time. Then, instead of gathering one observation at a time, we consider sequentially sampling k observations at a time in defining our proposed estimation strategies. We replace the customary sample standard deviation as an estimator for σ with a number of other pertinent estimators to come up with new and more appropriate stopping rules to suit the occasion. We do so because in real life we know that packaged items purchased in bulk often cost less per unit sample than the cost of an individual item. This article builds the whole array of estimation methodologies in order to address both FWCI and MRPE problems with appropriate first-order and second-order asymptotic analyses. These are followed by extensive sets of carefully laid out data analyses assisted via large-scale computer simulations. These are wrapped up with illustrations using breast cancer data. [ABSTRACT FROM AUTHOR]

We consider a problem of sequential testing a simple hypothesis against a simple alternative, based on observations of a discrete-time stochastic process X 1 , X 2 , ... , in the presence of a random horizon H. At any time n of the experiment, the statistician is only informed whether H > n or not. In this latter case, the experiment should be terminated and the final decision on the acceptance or rejection of the hypothesis should be taken on the basis of the available observations X 1 , X 2 , ... , X n ( n = 1 , 2 , ... ). H is assumed to be independent of the observations, and its distribution is known to the statistician. Under the random horizon, we consider a variant of the modified Kiefer-Weiss problem: given restrictions on the probabilities of errors, minimize the average sample size calculated under the assumption that the observations follow a fixed distribution, not necessarily one of those hypothesized. Under suitable conditions on the process and/or the horizon, we characterize the structure of all optimal sequential tests in this problem. Then, we apply these results to characterize optimal tests in the case of independent observations. On the basis of the general theory, more specific results are obtained for independent and identically distributed (i.i.d.) observations with a geometrically distributed horizon. In a simple sampling model, we solve the Kiefer-Weiss problem under the random horizon model. We also discuss the questions of Wald-Wolfowitz optimality in the presence of the random horizon. In particular, we show that the stopping rules of the optimal tests, minimizing the average sample size under one of the hypotheses, are randomized versions of those of Wald's sequential probability ratio tests. [ABSTRACT FROM AUTHOR]

The problem of quickest moving anomaly detection in networks is studied. Initially, the observations are generated according to a prechange distribution. At some unknown but deterministic time, an anomaly emerges in the network. At each time instant, one node is affected by the anomaly and receives data from a post-change distribution. The anomaly moves across the network, and the node that it affects changes with time. However, the trajectory of the moving anomaly is assumed to be unknown. A discrete-time Markov chain is employed to model the unknown trajectory of the moving anomaly in the network. A windowed generalized likelihood ratio–based test is constructed and is shown to be asymptotically optimal. Other detection algorithms including the dynamic Shiryaev-Roberts test, a quickest change detection algorithm with recursive change point estimation, and a mixture cumulative sum (CUSUM) algorithm are also developed for this problem. Lower bounds on the mean time to false alarm are developed. Numerical results are further provided to compare their performances. [ABSTRACT FROM AUTHOR]

Quick detection of common changes is critical in sequential monitoring of multistream data where a common change is a change that only occurs in a portion of panels. After a common change is detected by using a combined cumulative sum Shiryaev-Roberts (CUSUM-SR) procedure, we first study the joint distribution for values of the CUSUM process and the estimated delay detection time for the unchanged panels. A Benjamini-Hochberg (BH) method using the asymptotic exponential property for the CUSUM process is developed to isolate the changed panels with control on the false discovery rate (FDR). The common change point is then estimated based on the isolated changed panels. Simulation results show that the proposed method can also control the false non-discovery rate (FNR) by properly selecting the FDR. [ABSTRACT FROM AUTHOR]

We have designed a sequential minimum risk point estimation (MRPE) strategy for the unknown mean of a normal population having its variance unknown too. This is developed under a Linex loss plus linear cost of sampling. A number of important asymptotic first-order and asymptotic second-order properties' characteristics have been developed and proved thoroughly. Extensive sets of simulations tend to validate nearly all of these asymptotic properties for small to medium to large optimal fixed sample sizes. [ABSTRACT FROM AUTHOR]

In this paper, we develop appropriate sampling methodologies for testing hypotheses regarding the difference of mean values from two independent (or dependent) normal populations when their variances are unknown and unequal. We design two-stage and purely sequential testing methodologies of hypotheses for comparing the unknown means by determining the appropriate sample sizes while controlling both type-I and type-II error probabilities at or below preassigned levels α, β respectively. Such methodologies are constructed under both unequal and equal sample size designs. We prove that both two-stage and purely sequential testing strategies enjoy a number of practically appealing properties. Extensive sets of computer simulations and real data analyses empirically validate our theoretical findings. [ABSTRACT FROM AUTHOR]

This work deals with construction of fixed accuracy confidence intervals for the common variance in compound symmetric multivariate normal distribution by adopting a Stein-type two-stage setup and a completely sequential setup. Related asymptotics are developed and the procedures are evaluated by simulation study. [ABSTRACT FROM AUTHOR]

This article provides a two-stage procedure to develop a fixed-width confidence interval of log odds ratio in a joint binomial and inverse binomial setting where the stopping rule is obtained by adopting a two-stage procedure on the number of index subjects for the inverse sampling. A purely sequential version of this procedure is also studied. Different asymptotic results associated with the procedures are obtained. The findings are supported by detailed simulation study followed by one data example. [ABSTRACT FROM PUBLISHER]

Universal outlier hypothesis testing is studied in a sequential setting. Multiple observation sequences are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are generated by an “outlier” distribution, distinct from a common “typical” distribution governing the majority of the sequences. Apart from being distinct, the outlier and typical distributions can be arbitrarily close. The goal is to design a universal test to best discern all the outlier sequences. A universal test with the flavor of the repeated significance test is proposed and its asymptotic performance, as the error probability goes to zero, is characterized under various universal settings. The proposed test is shown to be universally consistent. For the model with at most one outlier, conditioned on the outlier being present, the test is shown to be asymptotically optimal universally when the typical distribution is known and as the number of sequences goes to infinity when neither the outlier nor the typical distribution is known. With multiple identical outliers, the test is shown to be asymptotically optimal universally when the number of outliers is the largest possible and with the typical distribution being known, and its asymptotic performance with neither the outlier nor the typical distribution being known is also characterized. Extensions of the findings to models with multiple distinct outliers are also discussed. In all cases, it is shown that the asymptotic performance guarantees for the proposed test when neither the outlier nor the typical distribution is known converge to those when the typical distribution is known as the number of sequences goes to infinity. [ABSTRACT FROM PUBLISHER]

The Levin-Robbins-Leu (LRL) family of sequential subset selectionprocedures admits of a simple formula that provides a lower bound for the probability of various types of subset selection events. Here we demonstrate that a corresponding lower bound formula holds forlattice eventswhen using the nonadaptive family member with binary outcomes. Lattice events are more general than the type of events that we previously considered in demonstrating that the nonadaptive LRL procedure selects “acceptable” subsets with an arbitrarily large prespecified probability irrespective of the true population parameters. Interestingly, the proof of the lower bound formula for lattice events that we present here simplifies the methods previously used and sheds additional light on why the lower bound formula holds. As for acceptable subset selection, we conjecture that the other LRL family members, which allow for adaptive elimination of inferior candidates and/or recruitment of superior candidates, obey the same lower bound formulas for lattice events. [ABSTRACT FROM AUTHOR]

The quickest multidecision change detection/isolation problem is the generalization of the quickest changepoint detection problem to the case ofMpost-change hypotheses. It is necessary to detect the change in distribution as soon as possible and to indicate which hypothesis is true after a change occurs. Both the rate of false alarms and the misidentification (false isolation) rate should be controlled by given levels. Several detection/isolation procedures that asymptotically minimize the tradeoff between the expected detection delay and the false alarm/false isolation rates in the worst-case scenario are discussed. [ABSTRACT FROM AUTHOR]

Letbe a sequence of consistent estimators of a parameter Δ in some response-adaptive design, where the data are not independent and identically distributed. A fixed-width confidence interval estimation involves determining a positive integer ν such that, for a givend(> 0) and α ∈ (0, 1), the coverage probability of the random intervalfor Δ is at least (1 − α). In this article, we first investigate such confidence intervals for a general structure of response-adaptive allocation design, for two-treatment allocation in clinical trials, in a complete sequential setup. We consider some examples in this context. We then carry out this exercise for a Stein-type two-stage procedure, where the parameters are estimated based on the first-stage data, which are then used for the allocation in the second stage. Fixed-width confidence intervals are obtained in this context. The procedures are evaluated by simulations. We then illustrate the proposed methodology using some real data set. [ABSTRACT FROM PUBLISHER]

There is an extensive literature on fixed-size confidence regions for the regression parameters in a linear model withpregressors, attaining a prescribed coverage probability whenpis fixed and the sizedapproaches 0. Motivated by recent developments in regression modeling in response to applications for whichpis considerably larger than the sample size, we develop herein a more versatile sequential methodology for fixed-size confidence regions that can handle the casep = p(d) → ∞ asd → 0. [ABSTRACT FROM PUBLISHER]

Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. This result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function, with respect to these multipliers, coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is jointly linear in the cost function and the Lagrangian multipliers and can be solved for both with off-the-shelf algorithms. To illustrate the procedure, optimal sequential tests for Gaussian random sequences with different dependency structures are derived, including the Gaussian AR(1) process. [ABSTRACT FROM AUTHOR]

A sensor network is considered where a sequence of random variables is observed at each sensor. At each time step, a processed version of the observations is transmitted from the sensors to a common node called the fusion center. At some unknown point in time the distribution of the observations at all of the sensor nodes changes. The objective is to detect this change in distribution as quickly as possible, subject to constraints on the false alarm rate and the cost of observations taken at each sensor. Minimax problem formulations are proposed for the above problem. A data-efficient algorithm is proposed in which an adaptive sampling strategy is used at each sensor to control the cost of observations used before change. To conserve the cost of communication an occasional binary digit is transmitted from each sensor to the fusion center. It is shown that the proposed algorithm is globally asymptotically optimal for the proposed formulations, as the false alarm rate goes to zero. [ABSTRACT FROM AUTHOR]

A new model for controlled sensing for multihypothesis testing is proposed and studied in the sequential setting. This new model, termed acontrolled Markovian observationmodel, exhibits a more complicated memory structure in the controlled observations than existing models. In addition, instead of penalizing just the delay until the final decision time as in standard sequential hypothesis testing problems, a much more general cost structure is considered that entails accumulating the total control cost with respect to an arbitrary control cost function. An asymptotically optimal test is proposed for this new model and is shown to satisfy anoptimalitycondition formulated in terms of decision-making risk. It is shown that the optimal causal control policy for the controlled sensing problem is self-tuning, in the sense of maximizing an inherent “inferential” reward simultaneously under every hypothesis, with the maximal value being the best possible corresponding to the case where the true hypothesis is known at the outset. Another test is also proposed to meetdistinctly predefinedconstraints on the various decision risksnonasymptotically, while retaining asymptotic optimality. [ABSTRACT FROM PUBLISHER]