On the empirical Bayes approach to multiple decision problems with sequential components

The empirical Bayes approach to multiple decision problems with a sequential decision problem as the component is studied. An empirical Bayesm-truncated sequential decision procedure is exhibited for general multiple decision problems. With a sequential component, an empirical Bayes sequential decision procedure selects both a stopping rule function and a terminal decision rule function for use in the component. Asymptotic results are presented for the convergence of the Bayes risk of the empirical Bayes sequential decision procedure.     

Multisource Bayesian sequential binary hypothesis testing problem

We consider the problem of testing two simple hypotheses about unknown local characteristics of several independent Brownian motions and compound Poisson processes. All of the processes may be observed simultaneously as long as desired before a final choice between hypotheses is made. The objective is to find a decision rule that identifies the correct hypothesis and strikes the optimal balance between the expected costs of sampling and choosing the wrong hypothesis. Previous work on Bayesian sequential hypothesis testing in continuous time provides a solution when the characteristics of these processes are tested separately. However, the decision of an observer can improve greatly if multiple information sources are available both in the form of continuously changing signals (Brownian motions) and marked count data (compound Poisson processes). In this paper, we combine and extend those previous efforts by considering the problem in its multisource setting. We identify a Bayes optimal rule by solving an optimal stopping problem for the likelihood-ratio process. Here, the likelihood-ratio process is a jump-diffusion, and the solution of the optimal stopping problem admits a two-sided stopping region. Therefore, instead of using the variational arguments (and smooth-fit principles) directly, we solve the problem by patching the solutions of a sequence of optimal stopping problems for the pure diffusion part of the likelihood-ratio process. We also provide a numerical algorithm and illustrate it on several examples.     

Bayes theorem,information number and behavior of posteior distributions

Summary According to the sequential decision theory, one is directed to; (i) stop when the posterior distribution, having been successively watched, frist enters the domain in which by stopping he loses less, and then (ii) take that action which gives him the least loss averaged w.r.t. the current posterior distribution (Blackwell-Girshick [1], p. 254). For some reasons we don't know, however, the successive behavior of posterior distribution never draws the full attention of statisticians and probabilists, though some of them have developed the asymptotic theory. As far as the sequential decision theory is concerned, however, the asymptotic behavior tells us nothing, since the stopping rule is only related to finite time. Toe compute the posterior distribution time by time is a laborious task. But to site all possible posterior distributions sometimes facilitates our manipulation of the sequential decision problem. This paper proposes some methods. The Institute of Statistical Mathematics     

A sequential test for the failure rate of an exponential distribution with censored data

This paper provides an optimal sequential decision procedure for deciding between two composite hypotheses about the unknown failure rate of an exponential distribution, using censored data. The procedure has two components, a stopping time and a decision function. The optimal stopping time minimizes the expected total loss due to a wrong decision plus cost of observing the process. The optimal decision function is easily characterized once a stopping time has been specified. The main result determines the continuation region for the optimal decision procedure     

Asymptotic Optimality of Certain Multihypothesis Sequential Tests: Non‐i.i.d. Case

It is known that certain combinations of one‐sided sequential probability ratio tests are asymptotically optimal (relative to the expected sample size) for problems involving a finite number of possible distributions when probabilities of errors tend to zero and observations are independent and identically distributed according to one of the underlying distributions. The objective of this paper is to show that two specific constructions of sequential tests asymptotically minimize not only the expected time of observation but also any positive moment of the stopping time distribution under fairly general conditions for a finite number of simple hypotheses. This result appears to be true for general statistical models which include correlated and non‐homogeneous processes observed either in discrete or continuous time. For statistical problems with nuisance parameters, we consider invariant sequential tests and show that the same result is valid for this case. Finally, we apply general results to the solution of several particular problems such as a multi‐sample slippage problem for correlated Gaussian processes and for statistical models with nuisance parameters. This revised version was published online in June 2006 with corrections to the Cover Date.     

几何分布时间序贯检验的贝叶斯推断

设有统计模型｛ｘ,Ｂｘ,Ｐθ｝,θ∈（０,１）,其中Ｐθ为几何分布：Ｐθ（Ｘ＝ｋ）＝（１－θ）θ＾ｋ－１ｋ＝１,２,…。考虑检验问题：θ＝θｏ ｖｓ． θ＝θ１（０〈θ０〈θ１〈１）本文对一种依次试验的时间序贯样本,给出了上述检验问题的贝叶斯停止判决法则,其中损失函数为试验费用和误判损失之和,贝叶斯停止判决法则由后验概率的两组界（上界和下界）所给出。     

Optimal decision under ambiguity for diffusion processes

In this paper we consider stochastic optimization problems for an ambiguity averse decision maker who is uncertain about the parameters of the underlying process. In a first part we consider problems of optimal stopping under drift ambiguity for one-dimensional diffusion processes. Analogously to the case of ordinary optimal stopping problems for one-dimensional Brownian motions we reduce the problem to the geometric problem of finding the smallest majorant of the reward function in a two-parameter function space. In a second part we solve optimal stopping problems when the underlying process may crash down. These problems are reduced to one optimal stopping problem and one Dynkin game. Examples are discussed.     

The monotone case approach for the solution of certain multidimensional optimal stopping problems

This paper studies explicitly solvable multidimensional optimal stopping problems of sum- and product-type in discrete and continuous time using the monotone case approach. It gives a review on monotone case stopping using the Doob decomposition, resp. Doob–Meyer decomposition in continuous time, also in its multiplicative versions. The approach via these decompositions leads to explicit solutions for a variety of examples, including multidimensional versions of the house-selling and burglar’s problem, the Poisson disorder problem, and an optimal investment problem.     

A general method for finding the optimal threshold in discrete time

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem. The optimal threshold is given by the threshold of the myopic stopping time in the auxiliary problem. Supplying also a sufficient condition for our approach to work, we obtain solutions for many prominent examples in the literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev, and the American put in option pricing under general conditions. As a further application we show that for underlying random walks (and Lévy processes in continuous time), general monotone and log-concave reward functions g lead to one-sided stopping problems.     

A partially observable decision problem under a shifted likelihood ratio ordering

In this paper, we discuss a partially observable sequential decision problem under a shifted likelihood ratio ordering. Since we employ the Bayes' theorem for the learning procedure, we treat this problem under several assumptions. Under these assumptions, we obtain some fundamental results about the relation between prior and posterior information. We also consider an optimal stopping problem for this partially observable Markov decision process.     

The Wiener Sequential Testing Problem with Finite Horizon

We present a solution of the Bayesian problem of sequential testing of two simple hypotheses about the mean value of an observed Wiener process on the time interval with finite horizon. The method of proof is based on reducing the initial optimal stopping problem to a parabolic free-boundary problem where the continuation region is determined by two continuous curved boundaries. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundaries can be characterized as a unique solution of the coupled system of two nonlinear integral equations.     

Empirical Bayes sequential estimation for exponential families: The untruncated component

We consider the empirical Bayes decision problem where the component problem is the sequential estimation of the mean of one-parameter exponential family of distributions with squared error loss for the estimation error and a cost c>0 for each observation. The present paper studies the untruncated sequential component case. In particular, an untruncated asymptotically pointwise optimal sequential procedure is employed as the component. With sequential components, an empirical Bayes decision procedure selects both a stopping time and a terminal decision rule for use in the component with parameter . The goodness of the empirical Bayes sequential procedure is measured by comparing the asymptotic behavior of its Bayes risk with that of the component procedure as the number of past data increases to infinity. Asymptotic risk equivalence of the proposed empirical Bayes sequential procedure to the component procedure is demonstrated.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant GP7987.     

Existence and explicit determination of optimal stopping times

We consider large classes of continuous time optimal stopping problems for which we establish the existence and form of the optimal stopping times. These optimal times are then used to find approximate optimal solutions for a class of discrete time problems.     

Sequential Fixed-Width Confidence Interval for the Product of Two Means

For the product of two population means, the problem of constructing a fixed-width confidence interval with preassigned coverage probability is considered. It is shown that the optimal sample sizes which minimize the total sample size and at the same time guarantee a fixed-width confidence interval of desired coverage depend on the unknown parameters. In order to overcome this, a fully sequential procedure consisting of a sampling scheme and a stopping rule are proposed. It is then shown that the sequential confidence interval is asymptotically consistent and the stopping rule is asymptotically efficient, as the width goes to zero. Furthermore, a second order result for the difference between the expected stopping time and the (total) optimal fixed sample size is established. The theoretical results are supported by appropriate simulations.     

A Gridding Method for Bayesian Sequential Decision Problems

This article introduces a numerical method for finding optimal or approximately optimal decision rules and corresponding expected losses in Bayesian sequential decision problems. The method, based on the classical backward induction method, constructs a grid approximation to the expected loss at each decision time, viewed as a function of certain statistics of the posterior distribution of the parameter of interest. In contrast with most existing techniques, this method has a computation time which is linear in the number of stages in the sequential problem. It can also be applied to problems with insufficient statistics for the parameters of interest. Furthermore, it is well-suited to be implemented using parallel processors.     

Optimal stopping with continuous control of piecewise deterministic Markov processes

In this paper we consider the problem of optimal stopping and continuous control on some local parameters of a piecewise-deterministic Markov processes (PDP's). Optimality equations are obtained in terms of a set of variational inequalities as well as on the first jump time operator of the PDP. It is shown that if the final cost function is absolutely continuous along trajectories then so is the value function of the optimal stopping problem with continuous control. These results unify and generalize previous ones in the current literature.     

Optimal Stochastic Impulse Control with Delayed Reaction

We study impulse control problems of jump diffusions with delayed reaction. This means that there is a delay δ>0 between the time when a decision for intervention is taken and the time when the intervention is actually carried out. We show that under certain conditions this problem can be transformed into a sequence of iterated no-delay optimal stopping problems and there is an explicit relation between the solutions of these two problems. The results are illustrated by an example where the problem is to find the optimal times to increase the production capacity of a firm, assuming that there are transaction costs with each new order and the increase takes place δ time units after the (irreversible) order has been placed.     

On when to stop sampling for the maximum

Suppose a sequential sample is taken from an unknown discrete probability distribution on an unknown range of integers, in an effort to sample its maximum. A crucial issue is an appropriate stopping rude determining when to terminate the sampling process. We approach this problem from a Bayesian perspective, and derive stopping rules that minimize loss functions which assign a loss to the sample size and to the deviation between the maximum in the sample and the true (unknown) maximum. We will show that our rules offer an extremely simple approximate solution to the well-known problem to terminate the Multistart method for continuous global optimization.     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

线性指数危险率模型的贝叶斯判别分析

设有两个总体Π0和Π1，其危险率为具有不同参数的线性函数。对于待观测的寿命样本X，给出了相应的判别分析问题的Bayes停止判决法则，其中损失函数包括试验费用和误判损失两部分。 