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We analyze the article by Han and Tsung (2009) “The Optimal Stopping Time for Detecting Changes in Discrete Time Markov Processes,” and demonstrate that it is seriously flawed.

In this paper we prove that Page's stopping time (CUSUM test) is optimal for detecting the changes in Markov processes that have invariant distribution.

Exact reliability analysis for complex repairable systems are usually difficult because of the complicated failure process that can result from repair policy. A common approach in practice is to use a simplified process such as the Power Law process which, although not exact, yields useful practical results. In this paper we consider the problem of determining a stopping time when estimating parameters of a Power Law process. The Bayes method is used in order to obtain an optimal time.

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 formula...

Optimal strategies are known for the finite and infinite horizon discrete-time search with constant unit cost and without recall. These strategies were obtained in the theory of optimal stopping, b...

Stein (1945) proposed a two-stage sequential methodology of inference that influenced numerous areas of statistics. In this article, the Stein's methodology is used and expands upon nonparametric estimation of the directional probability density. The aim is to propose a data-driven nonparametric sequential procedure that mimics performance of an oracle that knows smoothness of an estimated density and minimizes the mean stopping time given an assigned mean integrated squared error. For such a setting, using a sequential estimator is the must because smoothness of the density is unknown. It is known that for a general random variable the stated problem has no solution because an estimator cannot perform as well as the minimax oracle. At the same time, this article shows that for the case of directional density, under a mild assumption, there exists a data-driven two-stage sequential procedure that is minimax and adapts to unknown smoothness of an underlying density. Furthermore, we are able to solv...

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.

Optimal stopping theory is a key part of sequential analysis and applied probability. Dr. Soren Christensen’s article is an important contribution to solving a class of optimal stopping problems for Markov processes mainly in continuous time, but a certain expansion to discrete time is also given. This discussion contains several issues that naturally arise in many statistical applications.

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.

Professors T. N. Sriram and Ross Iaci have considered the problem of sequential estimation of the autoregressive parameter β in an AR(1) model and construction of sequential confidence region for a parameter vector θ in a TAR(1) model. Using extensive simulations, the authors have attempted to resolve the theoretical conjuncture that a fully sequential stopping time can have a negative regret. Their work opens up new directions for further research and establishes the need for second-order expansions for the regret. In this discussion, I briefly address some issues related to the simulation studies for an AR(1) model.

We look at a Poisson process where the arrival rate changes at some unknown time point. We monitor this process only at certain time points. At each time point, we count the number of arrivals that happened in that time interval. In previous work, it was assumed that the time intervals were fixed in advance. We relax this assumption to assume that the time intervals in which the process are monitored is also random. For a loss function consisting of the cost of late detection and a penalty for early stopping, we develop, using dynamic programming, the one- and two-step look-ahead Bayesian stopping rules. We then compare various observation schemes to determine the best model. We provide some numerical results to illustrate the effectiveness of the detection procedures.

The problem of partitioning with respect to a control or a standard is an important statistical problem in the area of selection and ranking. In the last 60 + years a number of formulations have been proposed for this problem utilizing the two-stage sampling strategy of Stein (1945) and other multistage sampling strategies. One such formulation, which had generalized many other formulations, was proposed in Tong (1969) for the normally distributed populations. Since then the formulation in Tong (1969) has been used by a multitude of researchers and practitioners. The indifference zone in Tong's (1969) formulation is a region in which the experimenter is indifferent to differentiate any treatment mean from the control treatment mean. However, this indifference zone also serves the additional role of defining the boundaries for “bad” and “good” treatments compared to the control treatment. Though the formulation in Tong (1969) is straightforward and easy to implement, this additional role that the i...

This article presents a sequential estimation procedure for an unknown regression function. Observed regressors and noises of the model are supposed to be dependent and form sequences of dependent numbers. Two types of estimators are considered. Both estimators are constructed on the basis of Nadaraya-Watson kernel estimators. First, sequential estimators with given bias and mean square error are defined. According to the sequential approach the duration of observations is a special stopping time. Then on the basis of these estimators of a regression function, truncated sequential estimators on a time interval of a fixed length are constructed. At the same time, the variance of these estimators is controlled by a (non-asymptotic) bound. In addition to nonasymptotic properties, the limiting behavior of presented estimators is investigated. It is shown, in particular, that by the appropriate chosen bandwidths both estimators have optimal (as compared to the case of independent data) rates of convergence of Nadaraya-Watson kernel estimators. © 2013 Copyright Taylor and Francis Group, LLC. 32 3 243 266 Cited By :3

In this article we extend Shiryaev's quickest change detection formulation by also accounting for the cost of observations used before the change point. The observation cost is captured through the average number of observations used in the detection process before the change occurs. The objective is to select an on–off observation control policy that decides whether or not to take a given observation, along with the stopping time at which the change is declared, to minimize the average detection delay, subject to constraints on both the probability of false alarm and the observation cost. By considering a Lagrangian relaxation of the constraint problem and using dynamic programming arguments, we obtain an a posteriori probability-based two-threshold algorithm that is a generalized version of the classical Shiryaev algorithm. We provide an asymptotic analysis of the two-threshold algorithm and show that the algorithm is asymptotically optimal—that is, the performance of the two-threshold algorith...

In clinical trials, continuous monitoring is required when a large but nonsignificant test statistic is observed at an interim analysis. We consider a power family of continuous stopping boundaries that are optimal in the sense that they minimize the average stopping time for the given overall Type I error rate and power. The problem is formulated as a constrained optimization problem and is solved numerically using the differential evolution algorithm. The main results are the ready-to-use approximately optimal continuous stopping boundaries for a wide range of values of power and the most commonly used overall levels of significance.

We consider a change detection problem in which the arrival rate of a Poisson process changes suddenly at some unknown and unobservable disorder time. It is assumed that the prior distribution of the disorder time is known. The objective is to detect the disorder time with an online detection rule (a stopping time) in a way that balances the frequency of false alarms and detection delay. So far in the study of this problem, the prior distribution of the disorder time is taken to be exponential distribution for analytical tractability. Here, we will take the prior distribution to be a phase-type distribution, which is the distribution of the absorption time of a continuous time Markov chain with a finite state space. We find an optimal stopping rule for this general case. We illustrate our findings on two numerical examples.

This note concerns a problem of optimally stopping a nondegenerate, two-dimensional, geometric Brownian motion Q t = (x t ,y t ), with the goal of maximizing where the supremum is taken over the class of all stopping times  Q , with finite expectation, H:ℝ+2 → ℝ is a measurable function satisfying a certain growth condition, and c > 0 is a positive constant. It is proved that, under certain conditions, the maximal value Φ(.,.) is a logarithmic function, and the optimal stopping time τ∗

Let S n be the partial sum of i.i.d. random variables X 1,X 2,…, and let N be the usual CUSUM stopping time based on S n . Under suitable conditions we determine ψ(α,β) = E exp(α S N − β N), where β > 0 and α is a suitable number. The given formula can be used to study the distributional properties of N, S N , and S N − h. Because the CUSUM based on maxima is reducible to N, the formula can be used to obtain the distributional properties of the maximal process as well. Several examples are discussed, and certain applications are shown in the so called trading securities. The formulas can also be used to study the distributional properties of a symmetric two-sided CUSUM.

The question whether a time series behaves as a random walk or as a stationary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This article studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a Monitoring procedure given by a control chart; i.e., a stopping time, which is related to a well-known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process that implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study.

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.

This article concerns a characterization of a vector-valued optimal stopping problem with only two reward functions. Under certain conditions, we give an explicit characterization of the Pareto stopping times for the present problem via a scalar-valued double optimal stopping problem. The latter problem is a natural extension of the classical McDonald-Siegel optimal stopping problem with one stopping rule.

We develop the exact distribution of the stopping variable of a sequential procedure that was originally given by Robbins and Siegmund (1974). The stopping variable was designed for estimating the log-odds in a sequence of Bernoulli trials. Using our exact distribution of the stopping variable, we give explicit formulas for the expected value and mean squared error for the estimator of the odds at stopping. An alternative two-stage procedure is then given and some of its important characteristics are exactly evaluated. It is shown that if the probability of success p is not too small or too large, the two-stage procedure is nearly as efficient as the purely sequential procedure. The results of this paper are then applied for designing an appropriate stopping time in a reliability experiment for estimating the ratio of the mean time between failures of two independent systems with exponential lifetimes.

For estimation of the parameters of a stable autoregressive process AR(p) by the least squares method, the paper proposes to use a particular stopping rule that essentially depends on the behavior of the minimal eigenvalue of the observed Fisher information matrix. The upper bounds for the mean square error of estimation are derived in the cases of known and unknown variances of the noise. Asymptotic formulas for the mean of the stopping time are given in both cases. Recommended by T. N. Sriram

Analogous to the usual stopping time for fixed-width confidence interval estimation of a real-valued parameter θ(F), a stopping time for fixed-width confidence band estimation of, a statistical functional process RF is considered. Techniques for showing that the resulting sequential confidence region has the correct asymptotic coverage probability are illustrated in several examples. Sufficient conditions for asymptotic efficiency and normality of the stopping time are given. Some implications of using the stopping time in conjunction with a bootstrap confidence band procedure are investigated in the context of estimating a dstribution function.

A new approximation method is proposed for calculating the operating characteristic function and the average sample number function of the sequential probability ratio test. This method conditions on the value of the test statistic immediately before the stopping time to produce mathematical formulas for the characteristics. An expression for the characteristic function of the stopping time is also obtained by this method. The values obtained by the proposed method are compared to a standard numerical method for the case where the underlying distribution is normal.

We consider distributions of several first exit times of compound Poisson processes (CPPs) with positive jumps from a region that is bounded by two parallel boundaries. Recursive formulas of exact distributions are given for a CPP with absolutely continuous jumps. Non-recursive formulas are given for a CPP with discrete jumps, and are used as a numerical approximation for the absolutely continuous case. The problem has applications in queueing and risk theory as well as sequential testing.

For testing the drift of Brownian motion with equal error probabilities. We provided an asymptotic solution, within the class of symmetric triangular stopping boundaries, to the Kiefer-Weiss problem of minimizing the maximum expected stopping time. The resulting asymptotic minimax triangular test (AMTT) is a slight improvement over Lordenś2-SPRT.

Two-stage and sequential procedures are developed for fixed-width interval estimation of the parameter β in a gamma distribution G(α, β) when α is known. Exact properties are obtained for the two-stage procedure and some asymptotics and approximations are given for the operating characteristics of the sequential procedure and some numerical computations are included.

In the house-hunting problem, i.i.d. random variables, X 1, X 2,… are observed sequentially at a cost of c > 0 per observation. The problem is to choose a stopping rule, N, to maximize E(X N − Nc). If the X's have a finite second moment, the optimal stopping rule is N* = min {n ≥ 1: X n > V*}, where V* satisfies E(X − V*)+ = c. The statement of the problem and its solution requires only the first moment of the X n to be finite. Is a finite second moment really needed? In 1970, Herbert Robbins showed, assuming only a finite first moment, that the rule N* is optimal within the class of stopping rules, N, such that E(X N − Nc)− > −∞, but it is not clear that this restriction of the class of stopping rules is really required. In this article it is shown that this restriction is needed, but that if the expectation is replaced by a generalized expectation, N* is optimal out of all stopping rules assuming only first moments.

This work compares the performance of all existing 2-CUSUM stopping rules used in the problem of sequential detection of a change in the drift of a Brownian motion in the case of two-sided alternatives. As a performance measure, an extended Lorden criterion is used. According to this criterion, the optimal stopping rule is an equalizer rule. This paper compares the performance of the modified drift harmonic mean 2-CUSUM equalizer rules with the performance of the best classical 2-CUSUM equalizer rule whose threshold parameters are chosen so that equalization is achieved. This comparison is made possible through the derivation of a closed-form formula for the expected value of a general classical 2-CUSUM stopping rule.

A linear model is considered in which the design variables may be functions of previous responses and/or auxiliary randomisation. The model is observed successive times, where t is a stopping time, and interest lies in estimating the parameters of the model. Approximations are derived for the bias and variance of the maximum likelihood estimators of the parameters at time t. The derivations involve differentiating the fundamental identity of sequential analysis. The accuracy of the approximations is assessed by simulation for a multi-armed clinical trial model proposed by Coad (1995), two autoregressive models and the sequential design of Ford and Silvey (1980). Very weak expansions are used to justify the approximations.

It is shown that under certain conditions the matrix sequential probability ratio test (SPRT) and the combinations of "rejecting" SPRTs minimize all moments of the stopping time distribution in the problem of sequential testing of several simple hypotheses for nonhomogeneous processes when probabilities of errors tend to zero. We consider the general case of observation process with discrete or continuous time parameter and asymmetric (relative to probabilities of errors) classes of tests.

We consider a Brownian motion X(t) with drift θ and various linear stopping boundaries, including boundaries popular for sequential testing. We show that the stopping time T and the observed value X(T) are jointly complete for the drift. We then derive the UMVUE of the drift and of some functions of the drift, including the variance of the drift estimate. The estimates are also truncation-adaptable, with uniformly minimum variance among such estimates.

We discuss the performance of “closed-end” fluctuation tests, used to detect changes in the structural stability of a model, in the case where the time horizon is large and the observations do not possess high enough moments. It is demonstrated via a simulation study that, in the latter case, the choice of the boundary function should take into account the number of moments existing; otherwise the actual size of the test may exceed the nominal level. We suggest an alternative “open-end” fluctuation procedure and study its asymptotic behavior. It turns out that if an improper boundary function is chosen, then the null hypothesis may be rejected even if it is true. This means that an appropriate choice of the boundary function requires some prior information about the tail heaviness of the observations. This is different from the cumulative sum-based monitoring schemes suggested by Chu et al. (1996) and further studied by Horvath et al. (2004).

We look at a Poisson process where the arrival rate changes at some unknown integer. At each integer, we count the number of arrivals that happened in that time interval. We assume that the arrival rates before and after the change are unknown. For a loss function consisting of the cost of late detection and a penalty for early stopping, we develop, using dynamic programming, the one- and two-step look-ahead Bayesian stopping rules. We provide some numerical results to illustrate the effectiveness of the detection procedures.

This paper considers sequantial producers to construct fixed-width confidence intervals for some function θ of mean μ and variance σ2 of normal distribution.Consideration is devoted to θ=exp( μ + σ2/2 ) and θ=μ/σ.Nonlinear renewal theory is used to drive asymptotic expansion of expectation of the stopping time and the estimate as the width of confidence interval decreases to zero.An improvement of the coverage probability is also discussed.

This paper deals with the sequential point estimation of the autoregressive parameters in a multiple autoregressive model using the least squares estimator.The sequential estimator is shown to be asymptotically risk efficient under some regularity conditions.The asymptotic normality and uniform integrability of standardized stopping rule are established.This paper also contain a second order approximation to expected stopping time and an expression for regret of the above stopping rule under certain smoothness conditions.

We consider Brownian motion with drift and stopping boundaries: linear upper and lower boundaries, and possibly a vertical boundary at a truncation point, all under conditions assuring a finite stopping time. T. W. Anderson (Ann. Math. Statist. 31, 1960) derived formulas for the distributions of the stopped process along these boundaries and for the associated expected stopping times. We present simpler formulas, and briefer derivations.

In this paper the distribution of a concomitant statistic in a Gaussian sequential setting is characterized. It is shown that the distribution of such a statistic, conditional on the value of the stopping time, can be represented as a convolution of a normal random variable with a stochastic integral of the monitoring process of the trial. For continuous sequential designs the characteristic function of this stochastic integral is shown to satisfy an integral equation. The tools developed here are relevant for secondary inference problems in sequential clinical trials.

Let X j , j = 1,…, n be independent and identically distributed random variables. Like in the classical secretary problem, the optimal stopper only observes Y j = 1, if X j is a (relative) record, and Y j = 0, otherwise. The actual X j values are not revealed. The goal is to maximize the expected X value at which one stops. We show that the optimal number of observations one should skip before considering stopping depends heavily on the underlying distribution.

In sequential analysis, investigation of stopping rules is important, as they govern the sampling cost and derivation and accuracy of frequentist inference. We study stopping rules in sampling from a population comprised of an unknown number of classes where all classes are equally likely to occur in each selection. We adopt Blackwell's criterion for a “more informative experiment” to compare stopping rules in our context and derive certain complete class results, which provide some guidance for selecting a stopping rule. We show that it suffices to let the stopping probability, at any time, depend only on the number of selections and the number of discovered classes up to that time. A more informative stopping rule costs a higher expected sample size, and conversely, any given stopping rule can be improved with an increment in expected sample size. Admissibility within all stopping rules with a uniform upper bound on average sample size is also discussed. Any fixed-sample-size rule is shown to b...