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Sequential change-point analysis, which identifies a change of probability distribution in an infinite sequence of random observations, has important applications in many fields. A good method should detect a change point as soon as possible, and keep a low amount of false alarms. As one of the most popular methods, Shiryaev-Roberts (SR) procedure holds many optimalities. However, its implementation requires the pre-change and post-change distributions to be known, which is not achievable in practice. In this paper, we construct a nonparametric version of the SR procedure by embedding different versions of empirical likelihood, assuming two training samples, before and after change, are available for parameter estimations. Simulations are conducted to compare the performance of the proposed method with existing methods. The results show that when the underlying distribution is unknown, and training sample sizes are small, the proposed modified procedure shows advantage by giving a smaller delay of detection. [ABSTRACT FROM AUTHOR]

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]

We propose a two-stage sequential method for obtaining tandem-width confidence intervals for a Bernoulli proportion p. The term "tandem-width" refers to the fact that the half-width of the confidence interval is not fixed beforehand; it is instead required to satisfy two different half-width upper bounds, h0 and h1, depending on the (unknown) values of p. To tackle this problem, we first propose a simple but useful sequential method for obtaining fixed-width confidence intervals for p, whose stopping rule is based on the minimax estimator of p. We observe Bernoulli(p) trials sequentially, and for some fixed half-width h = h0 or h1, we develop a stopping time T such that the resulting confidence interval for p, [ ], covers the parameter with confidence at least where is the maximum likelihood estimator of p at time T. Furthermore, we derive theoretical properties of our proposed fixed-width and tandem-width methods and compare their performances with existing alternative sequential schemes. The proposed minimax-based fixed-width method performs similarly to alternative fixed-width methods, while being easier to implement in practice. In addition, the proposed tandem-width method produces effective savings in sample size compared to the fixed-width counterpart and provides excellent results for scientists to use when no prior knowledge of p is available. [ABSTRACT FROM AUTHOR]

Joint detection and estimation refers to deciding between two or more hypotheses and, depending on the test outcome, simultaneously estimating the unknown parameters of the underlying distribution. This problem is investigated in a sequential framework under mild assumptions on the underlying random process. We formulate an unconstrained sequential decision problem, whose cost function is the weighted sum of the expected run-length and the detection/estimation errors. Then, a strong connection between the derivatives of the cost function with respect to the weights, which can be interpreted as Lagrange multipliers, and the detection/estimation errors of the underlying scheme is shown. This property is used to characterize the solution of a closely related sequential decision problem, whose objective function is the expected run-length under constraints on the average detection/estimation errors. We show that the solution of the constrained problem coincides with the solution of the unconstrained problem with suitably chosen weights. These weights are characterized as the solution of a linear program, which can be solved using efficient off-the-shelf solvers. The theoretical results are illustrated with two example problems, for which optimal sequential schemes are designed numerically and whose performance is validated via Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

In sequential analysis, an experimenter may gather information regarding an unknown parameter by observing random samples in successive steps. We emphasize a number of specific parametric models under a variety of loss functions. The total number of observations collected at termination is a positive integer-valued random variable, customarily denoted by . The exact probability distribution of N is often hard to obtain. However, under a set of regulatory conditions, our standardized version of the stopping variable from Definition 2.1 in Section 2.2 would follow a normal distribution in the asymptotic sense. In this article, we first show how these regulatory conditions build upon one another in order to conclude the asymptotic normality of such standardized stopping variable, . We provide exploratory data analysis (EDA) via a number of interesting illustrations obtained through large-scale simulation studies. We demonstrate the broad applicability of purely sequential methodologies included in this article along with the appropriateness of our conclusions regarding the asymptotic normality of the standardized stopping variable as a practical guideline. [ABSTRACT FROM AUTHOR]

Sequential procedure is developed for minimum risk point estimation of the Lomax distribution under quadratic plus linear cost of sampling of the shape parameter. Secondorder approximations are derived for the expected sample size and regret associated with sequential procedure. [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]

Truncated sequential test procedures are proposed for testing the mean time between failures of a system with exponential life distribution. The exact distributions of the stopping times for the sequential tests are derived by investigating boundary crossing times of homogeneous Poisson processes. The exact formulas for the expected values of the stopping variables and the survival functions are derived. Moreover, the operation characteristic functions are derived exactly for the proposed sequential tests. [ABSTRACT FROM AUTHOR]

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 indifference zone plays could make the formulation somewhat undesirable to the practitioner. To illustrate, assume that in a clinical trial a drug with less than 10% improvement over the placebo is considered as a drug with insignificant improvement and the drug is termed a "bad treatment" and a drug with more than 30% improvement over the placebo is considered a drug with significant improvement and the drug is termed a "good treatment." In such a scenario, Tong's formulation would define the indifference zone to be between 10% and 30% improvement over placebo, and any treatment with an effect size of between 10% and 30% improvement can be partitioned either a good treatment or a bad treatment without any penalty. Such a wide indifference zone makes the partition obtained to be rather misleading as the treatments termed good treatment or bad treatment could include treatments with effect size between 10% and 30%. One could potentially reduce the size of the indifference zone in Tong's formulation, but this would then alter the definition of the bad and good treatments, which are generally provided by the experts in the area. In this article, a generalization of Tong's formulation is proposed that will partition the treatments in the indifference zone as a separate identifiable group without altering the definition of the good and bad treatments. It is shown that the formulation in Tong (1969) is a limiting case of the generalized partition formulation proposed in this article. A fixed sample procedure is constructed for the case when the common population variance is known. In addition, a purely sequential procedure is proposed for the unknown variance case. The first-order and second-order asymptotic properties are derived and verified using Monte Carlo simulation studies. An example is provided to illustrate an application of the proposed generalization. [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]

In this letter, we introduce a sequential version of universal hypothesis testing, where the goal of the test is not only to decide between the known null hypothesis and some other unknown alternative hypothesis, but also to use a stopping time to stop the test as soon as rejecting the null hypothesis. Motivated by the method of types in information theory, we establish the sequential test by Höeffding's universal test associated with a curved stopping boundary. It is proved that this sequential test uniformly asymptotically minimizes average sample size for any other sequential test. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: In clinical trials, a covariate-adjusted response-adaptive (CARA) design allows a subject newly entering a trial a better chance of being allocated to a superior treatment regimen based on cumulative information from previous subjects, and adjusts the allocation according to individual covariate information. Since this design allocates subjects sequentially, it is natural to apply a sequential method for estimating the treatment effect in order to make the data analysis more efficient. In this paper, we study the sequential estimation of treatment effect for a general CARA design. A stopping criterion is proposed such that the estimates satisfy a prescribed precision when the sampling is stopped. The properties of estimates and stopping time are obtained under the proposed stopping rule. In addition, we show that the asymptotic properties of the allocation function, under the proposed stopping rule, are the same as those obtained in the non-sequential/fixed sample size counterpart. We then illustrate the performance of the proposed procedure with some simulation results using logistic models. The properties, such as the coverage probability of treatment effect, correct allocation proportion and average sample size, for diverse combinations of initial sample sizes and tuning parameters in the utility function are discussed. [Copyright &y& Elsevier]

The point of departure of the present article is Anscombe's seminal 1952 paper on limit theorems for randomly indexed processes. We discuss the importance of this result and mention some of its impact, mainly on stopped random walks. The main aim of the article is to illustrate the beauty and efficiency of what will be called the stopped random walk (SRW) method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the optimal stopping problem with a possible compensated refusal of reward. We discuss functionals of exponential Brownian motion. The optimal stopping time is defined on the set of all finite stopping times. The functionals under consideration correspond to payments for standard American options. [ABSTRACT FROM AUTHOR]

The optimal stopping problem with incomplete data for the Kalman-Bucy scheme is reduced to the optimal stopping problem with complete data. In the case of a continuous gain function it is proven that price convergence has an order (ϵ1 +ϵ2) when small perturbation parameters ϵ1, ϵ2 of an observable process tend to zero. [ABSTRACT FROM AUTHOR]

Sequential methods have been used for many applications; especially, when fixed sample procedures are not possible and/or when 'early stopping' of sampling is beneficial for applications. At the same time, the issue of how to make correct inferences when measurement errors are present has drawn considerable attention from statisticians. In this paper, the problems of sequential estimation of generalized linear models when there are measurement errors in both adaptive and fixed design cases are studied. The proposed sequential procedure is proved to be asymptotically consistent and efficient in the sense of Chow and Robbins [Ann Math Stat 36(2):457-462, 1965] when measurement errors decay gradually as the number of sequentially selected design points increases. This assumption is useful in sequentially designed experiments, and can also be fulfilled in the case when replicate measurements are available. Some numerical studies based on a Rasch model and a logistic regression model are conducted to evaluate the performance of the proposed procedure. [ABSTRACT FROM AUTHOR]

Abstract: The standard approach in change-point theory is to base the statistical analysis on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. The present paper is a continuation of Gut and Steinebach [2002. Truncated sequential change-point detection based on renewal counting processes. Scand.J.Statist. 29, 693–719] the main point being that here we look in more detail into the behaviour of the relevant stopping times, in particular the time it takes from the actual change-point until the change is detected, more precisely, we prove asymptotics for stopping times under alternatives. [Copyright &y& Elsevier]

Common software release procedures based on statistical techniques try to optimize the trade-off between further testing costs and costs due to remaining errors. We propose new software release procedures where the aim is to certify with a certain confidence level that the software does not contain errors. The underlying model is a discrete time model similar to the geometric Moranda model. The decisions are based on a mix of classical and Bayesian approaches to sequential testing and do not require any assumption on the initial number of errors. [ABSTRACT FROM AUTHOR]

Abstract: Consider a Galton–Watson process with immigration. This paper studies the limits of the sequential estimator, proposed by [Sriram, T.N., Basawa, I.V., and Huggins, R.M., (1991). Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232–2243.] and the modified sequential estimator, proposed by [Shete, S., Sriram, T.N., 1998. Fixed precision estimator of the offspring mean in branching processes. Stochastic Process. Appl. 77, 17–33.]. [Sriram, T.N., Basawa, I.V., Huggins, R.M., 1991. Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232–2243.] proved that the sequential estimators are asymptotically normal in the subcritical and critical cases. In this paper it is proved that the sequential estimators are asymptotically normal in the supercritical case and that the limiting distributions of the modified estimators, after being properly standardized, are normal as well. [Copyright &y& Elsevier]

Robert A. Wijsman made fundamental contributions to the field of sequential analysis, especially in the area of invariant sequential tests and exponential boundedness of stopping times. This article provides an overview of the main themes and ideas of this impressive body of work, along with some personal remembrances from family members and colleagues. [ABSTRACT FROM AUTHOR]

The first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear bather is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties. [ABSTRACT FROM AUTHOR]

The central limit theorem for a normalized summation of random number of i.i.d. random variables is well known. In this paper we improve the central limit theorem by providing a two-term expansion for the distribution when the random number is the first time that a simple random walk exceeds a given level. Some numerical evidences are provided to show that this expansion is more accurate than the simple normality approximation for a specific problem considered. [ABSTRACT FROM AUTHOR]