A Linear Programming Approach to Sequential Hypothesis Testing

Abstract

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.     

Construction of an Optimal Sequential Plan for Testing a Treatment for an Adverse Effect

Abstract

Classical sequential procedures that collect a single observation at a time are often found impractical, expensive, and time consuming. Sequentially planned procedures, or simply sequential plans, extend and generalize the concepts of sequential analysis by allowing observations to be collected in groups of variable sizes. After every group, all of the previously collected data are used to determine the next course of action. An optimal (Bayes) sequential plan minimizes the (Bayes) risk function that combines the decision loss, observation (variable) cost, and group (fixed) cost. In general, determining the optimal sequential plan remains an open and challenging problem mainly because it requires risk optimization over a huge and rather unstructured set of all sequential plans. This article demonstrates how to obtain the optimal solution for a particular class of problems that may arise in testing a treatment for a rare but severe adverse effect. This solution is obtained by studying a number of properties of the Bayes sequential plan such as transitivity and monotonicity. This allows one to reduce the search to a small, manageable set of sequential plans within which the optimal plan can be calculated.     

Empirical bayes sequential estimation of the mean

We consider the empirical Bayes problem where the component problem is the sequential estimation of the mean of a distribution with squared error decision loss plus a sampling cost. An empirical Bayes sequential estimation procedure is exhibited which is asymptotically optimal. Asymptotic efficiency of the empirical Bayes stopping time sequence is also established. The performance of the proposed empirical Bayes procedure is studied with the help of a Monte Carlo study.     

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

Abstract

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.     

A sequential sampling plan for exponential distribution

In this article, a sequential variable sampling plan is studied. Suppose that the quality of an item in a batch is measured by a random variable with exponential distribution; its parameter is unknown having a gamma prior distribution. Then by using Bayesian approach and considering a Markov decision process, the optimality equations for the minimum total expected cost are formulated. We show that an optimal decision rule will have a control limit structure and monotonicity. A backward induction method is suggested that is a finite algorithm for the numerical solution of the sequential sampling plan.     

基于参数估计的动态系统过失误差侦破与识别

The detection and identification of gross errors, especially measurement bias, plays a vital role in data reconciliation for nonlinear dynamic systems. Although parameter estimation method has been proved to be a powerful tool for bias identification, without a reliable and efficient bias detection strategy, the method is limited in efficiency and cannot be applied widely. In this paper, a new bias detection strategy is constructed to detect the presence of measurement bias and its occurrence time. With the help of this strategy, the number of parameters to be estimated is greatly reduced, and sequential detections and iterations are also avoided. In addition, the number of decision variables of the optimization model is reduced, through which the influence of the parameters estimated is reduced. By incorporating the strategy into the parameter estimation model, a new methodology named IPEBD (Improved Parameter Estimation method with Bias Detection strategy) is constructed. Simulation studies on a continuous stirred tank reactor (CSTR) and the Tennessee Eastman (TE) problem show that IPEBD is efficient for eliminating random errors, measurement biases and outliers contained in dynamic process data.     

A Comparison of 2-CUSUM Stopping Rules for Quickest Detection of Two-Sided Alternatives in a Brownian Motion Model

Abstract

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.     

Sequential estimation of the autoregressive parameter in a first order autoregressive process

This paper considers the problem of sequential point estimation of the autoregressive parameter in a first order autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be asymtotically risk efficient as the cost of estimation error tends to infinity, under certain regularity conditions. Furthermore, nonlinear renewal theory is used to obtain a second order approximation to the expected stopping time. The asymptotic normality and uniform integrability of the standardized stopping time are also established.     

Optimal Sequential Tests for Two Simple Hypotheses

Abstract

A general problem of testing two simple hypotheses about the distribution of a discrete-time stochastic process is considered. The main goal is to minimize an average sample number over all sequential tests whose error probabilities do not exceed some prescribed levels. As a criterion of minimization, the average sample number under a third hypothesis is used (modified Kiefer–Weiss problem). For a class of sequential testing problems, the structure of optimal sequential tests is characterized. An application to the Kiefer–Weiss problem for discrete-time stochastic processes is proposed. As another application, the structure of Bayes sequential tests for two composite hypotheses, with a fixed cost per observation, is given. The results are also applied for finding optimal sequential tests for discrete-time Markov processes. In a particular case of testing two simple hypotheses about a location parameter of an autoregressive process of order 1, it is shown that the sequential probability ratio test has the Wald–Wolfowitz optimality property.     

BAYES SEQUENTIAL ESTIMATION OF POISSON MEAN UNDER A LINEX LOSS FUNCTION

This paper considers Bayes sequential estimation of the mean of a Poisson distribution using a LINEX loss function and a cost c > 0 for each observation. Under a Gamma prior distribution, it is shown that an asymptotically pointwise optimal rule is asymptotically non-deficient in the sense of Woodroofe (1981).     

Exact Risks of Sequential Point Estimators of the Exponential Parameter

Abstract

Under purely sequential sampling schemes, a theory is developed for the exact determination of the distributions of two classes of stopping variables (rules) in order to handle point estimation problems for the parametric functionals in an exponential distribution. Explicit formulae are derived for the expected value and risks of sequential estimators of the mean, failure rate, and reliability function of an exponential distribution. These are utilized to compare performances of several competing estimators of the mean and the failure rate.     

Bayesian Sequential Estimation of a Drift of Fractional Brownian Motion

Abstract

We solve explicitly a Bayesian sequential estimation problem for the drift parameter μ of a fractional Brownian motion under the assumptions that a prior density of μ is Gaussian and that a penalty function is quadratic or Dirac-delta. The optimal stopping time for this case is deterministic.     

On Two‐Stage Estimation of the Spectral Density with Assigned Risk in Presence of Missing Data

Estimation with assigned risk is a classical statistical problem, and the theory is well developed for the case of directly observed (no missing) data. In this article a more complicated problem of estimation of the spectral density in presence of missing data is considered. First, the corresponding theory of sequential estimation with minimal expected stopping time is developed. Then it is shown that a two‐stage estimator may be used and it yields the minimal stopping time. Sample size of the first stage may be deterministic and in order smaller than a minimal stopping time, and then the first stage defines the size of the second stage. Furthermore, the estimator adapts to unknown smoothness of an underlying spectral density and to an underlying missing mechanism.     

Multiple Optimality Properties of the Shewhart Test

Abstract

For the problem of sequential detection of changes, we adopt the probability maximizing approach in place of the classical minimization of the average detection delay and propose modified versions of the Shiryaev, Lorden, and Pollak performance measures. For these alternative formulations, we demonstrate that the optimum sequential detection scheme is the simple Shewhart rule. Interestingly, we can also solve problems that under the classical setup have been open for many years, as optimum change detection with time-varying observations or with multiple postchange probability measures. For the latter, we also offer the exact solution for Lorden's original setup involving average detection delays, for the case where the average false alarm period is within certain limits.     

Iterative improvement of parameter estimation for model migration by means of sequential experiments

Determining an optimal design for estimation of parameters of a class of complex models expected to be built at a minimum cost is a growing trend in science and engineering. We adopt a scale-bias adjustment migration strategy for integrating base and new models based on similar nature underlying processes. Further, we propose a Bayesian sequential algorithm for obtaining the statistically most informative data about the migrated model for use in parameter estimation. The benefits of the proposed strategy over traditional approaches presented in recent reported work are demonstrated using Monte Carlo simulations.     

Sequential Bonferroni Methods for Multiple Hypothesis Testing with Strong Control of Family-Wise Error Rates I and II

Abstract

Sequential procedures are developed for simultaneous testing of multiple hypotheses in sequential experiments. Proposed stopping rules and decision rules achieve strong control of both family-wise error rates I and II. The optimal procedure is sought that minimizes the expected sample size under these constraints. Bonferroni methods for multiple comparisons are extended to sequential setting and are shown to attain an approximately 50% reduction in the expected sample size compared with the earlier approaches. Asymptotically optimal procedures are derived under Pitman alternative.     

Bernoulli Sequential Estimation of the Size of a Finite Population

Abstract

The problem considered is that of unbiased estimation of the size (N) of a finite closed population under Capture-mark-Release-Recapture (CMRR) sequential sampling procedure. Borrowing ideas from the Bernoulli sequential estimation and using the notions of “closed” and “pushed-up” sampling plans, we provide here an unified approach to the problem of unbiased estimation of N and, in particular, give a necessary and sufficient condition for unbiased estimability under an arbitrary stopping rule. The ideas are illustrated with several examples.     

基于O3/UV法在线COD检测的气体溶解量估计方法

针对臭氧协同紫外方法(O₃/UV)检测化学需氧量(COD)时存在溶解性气体影响测量精度的问题，提出了一种COD检测过程中气体溶解量的估计方法，用于对COD检测模型的补偿。采集不同浓度的COD标准水样在消解过程中的测量数据和实验分析数据，基于PLS-LSSVMs建立溶解氧量和溶解二氧化碳量的估计模型，将模型的输出作为COD检测模型的补偿项。实验结果表明，基于PLS-LSSVMs建立的模型比PLS或者LSSVMs单独建立的模型估计精度高。采用溶解气体量估计模型进行补偿后的O₃/UV法检测COD与国标法测量结果相对误差均小于5%。对提高O₃/UV法检测COD精度具有重要意义。     

Decision-theoretical formulation of the calibration problem

The choice of calibration policy is of basic importance in analytical chemistry. A prototype of the practical calibration problem is formulated as a mathematical task and a Bayesian solution of the resulting decision problem is presented. The optimum feedback calibration policy can then be found by dynamic programming. The underlying parameter estimation and filtering are solved by updating relevant conditional distributions. In this way: the necessary information is specified (for instance, the need for knowledge of the probability distribution of unknown samples is clearly recognized as the conceptually unavoidable informational source); the relationship of the information gained from a calibration experiment to the ultimate goal of calibration, i.e., to the estimation of unknown samples, is explained; an ideal solution is given which can serve for comparing various ways of calibration; and a consistent and conceptually simple guideline is given for using decision theory when solving problems of analytical chemistry containing uncertain data. The abstract formulation is systematically illustrated by an example taken from gas chromatography.     

On the Worth of Perfect Information in Bandits with Random Discounting

Abstract

A discrete-time bandit process is a sequential decision problem in which one selects from a finite number of stochastic processes at each stage, and receives as a reward the product of the value of the observed process and a discount factor. In the classical formulation of the problem, the sequence of discount factors is known in advance, but the distributions governing the observations on the stochastic processes are not known. In this paper we consider the case in which the discount sequence is random and compare the reward of observers who have information about the discount sequence from the start with those who do not. In particular, we obtain bounds on the expected rewards for processes in which the discount sequence is random.