用于连续抽样方案的两类更优的中止规则

本文提出并讨论了适用于各类连续抽样方案的两参数中止规则 [N ,c]和 [R ,d]。作为应用 ,以MIL STD 12 35B中公布的CSP V方案为例 ,计算并给出了相应的中止参数 ,还将 [N ,c],[R ,d]与已有的两类规则 [S],[R]进行了特性比较。结果表明 ,[N ,c]和 [R ,d]均具有更优的统计特性     

Additive comparisons of stopping values and supremum values for finite stage multiparameter stochastic processes

This paper concerns the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. In the classical optimal stopping problems, the comparisons between the expected reward of a player with complete foresight and the expected reward of a player using nonanticipating stop rules, known as prophet inequalities, have been studied by many authors. Ratio comparisons between these values in the case of multiparameter optimal stopping problems are studied by Krengel and Sucheston (1981) [9] and Tanaka (2007, 2006) [14] and [15]. In this paper an additive comparison in the case of finite stage multiparameter optimal stopping problems is given.     

Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.     

Sequential stopping rules for the multistart algorithm in global optimisation

In this paper a sequential stopping rule is developed for the Multistart algorithm. A statistical model for the values of the observed local maxima of an objective function is introduced in the framework of Bayesian non-parametric statistics. A suitablea-priori distribution is proposed which is general enough and which leads to computationally manageable expressions for thea-posteriori distribution. Sequential stopping rules of thek-step look-ahead kind are then explicitly derived, and their numerical effectiveness compared.     

Practical stopping criteria for validating safety-critical software by estimating impartial reliability

In this research, we investigate stopping rules for software testing and propose two stopping rules from the aspect of software reliability testing based on the impartial reliability model. The impartial reliability difference (IRD-MP) rule considers the difference between the impartial transition-probability reliabilities estimated for both software developer and consumers at their predetermined prior information levels. The empirical–impartial reliability difference (EIRD-MP) rule suggests stopping a software test when the computed empirical transition reliability is tending to its estimated impartial transition reliability. To insure the high-standard requirement for safety-critical software, both rules take the maximum probability (MP) of untested paths into account.     

Exit criteria and monotonicity in compound quadrature of Gaussian type

Summary In this short note, for compound quadrature rules of Gaussian type, we prove stopping rules and monotonicity results based on Peano-kernel methods.     

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.     

Descentwise inexact proximal algorithms for smooth optimization

The proximal method is a standard regularization approach in optimization. Practical implementations of this algorithm require (i)?an algorithm to compute the proximal point, (ii)?a rule to stop this algorithm, (iii)?an update formula for the proximal parameter. In this work we focus on?(ii), when smoothness is present??so that Newton-like methods can be used for?(i): we aim at giving adequate stopping rules to reach overall efficiency of the method. Roughly speaking, usual rules consist in stopping inner iterations when the current iterate is close to the proximal point. By contrast, we use the standard paradigm of numerical optimization: the basis for our stopping test is a ??sufficient?? decrease of the objective function, namely a fraction of the ideal decrease. We establish convergence of the algorithm thus obtained and we illustrate it on some ill-conditioned problems. The experiments show that combining the proposed inexact proximal scheme with a standard smooth optimization algorithm improves the numerical behaviour of the latter for those ill-conditioned problems.     

Sequential Estimation of the Maximum in a Model for Corrosion Data

One method of monitoring corrosion in an underground storage tank involves placing a sensor in the tank and running it around the tank's interior. As it runs, the sensor records the local thickness of the tank. In this paper we consider the problem of estimating the maximum pit depth by providing a confidence interval that achieves both a specified confidence level and a specified degree of precision. A particular model, the three-parameter beta, is considered, and a stopping rule for determining the sample size is proposed. It is shown that the stopping rule achieves the desired confidence level and precision, asymptotically as the precision requirement becomes increasingly stringent. Moreover, the stopping rule is asymptotically efficient in terms of sample size. The limiting distribution of the stopping rule is derived, and simulation results are presented to supplement the asymptotics with finite sample size behavior.     

Structured policies in the sequential design of experiments

A general control model under uncertainty is considered. Using a Bayesian approach and dynamic programming, we investigate structural properties of optimal decision rules. In particular, we show the monotonicity of the total expected reward and of the so-called Gittins-Index. We extend the stopping rule and the stay-on-a-winner rule, which are well-known in bandit problems. Our approach is based on the multivariate likelihood ratio order andTP ₂ functions.     

A leavable bounded-velocity stochastic control problem

This paper studies bounded-velocity control of a Brownian motion when discretionary stopping, or ‘leaving’, is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of distance from the origin. There are two versions of this problem: the fully observed case, in which the control multiplies a known gain, and the partially observed case, in which the gain is random and unknown. Without the extra feature of stopping, the fully observed problem originates with Beneš (Stochastic Process. Appl. 2 (1974) 127–140), who showed that the optimal control takes the ‘bang–bang’ form of pushing with maximum velocity toward the origin. We show here that this same control is optimal in the case of discretionary stopping; in the case of power-law costs, we solve the variational equation for the value function and explicitly determine the optimal stopping policy.We also discuss qualitative features of the solution for more general cost structures. When no discretionary stopping is allowed, the partially observed case has been solved by Beneš et al. (Stochastics Monographs, Vol. 5, Gordon & Breach, New York and London, pp. 121–156) and Karatzas and Ocone (Stochastic Anal. Appl. 11 (1993) 569–605). When stopping is allowed, we obtain lower bounds on the optimal stopping region using stopping regions of related, fully observed problems.     

Stopping rules,permutation invariance and sufficiency principle

In the context of sequential (point as well as interval) estimation, a general formulation of permutation-invariant stopping rules is considered. These stopping rules lead to savings in the ASN at the cost of some elevation of the associated risk—a phenomenon which may be attributed to the violation of the sufficiency principle. For the (point and interval) sequential estimation of the mean of a normal distribution, it is shown that such permutation-invariant stopping rules may lead to a substantial saving in the ASN with only a small increase in the associated risk.Work partially supported by (i) Office of Naval Research, Contract Number N00014-85-K-0548, and (ii) Office of Naval Research, Contract Number N00014-83-K-0387.     

Adaptive Importance Sampling for simulating copula-based distributions

In this paper, we propose a generalization of importance sampling, called Adaptive Importance Sampling, to approximate simulation of copula-based distributions. Unlike existing methods for copula simulation that have appeared in the literature, this algorithm is broad enough to be used for any absolutely continuous copula. We provide details of the algorithm including rules for stopping the iterative process and consequently assess its performance using extensive Monte Carlo experiments. To assist in its extension to several dimensions, we discuss procedures for identifying the crucial parameters in order to achieve desirable results especially as the size of the dimension increases. Finally, for practical illustration, we demonstrate the use of the algorithm to price First-to-Default credit swap, an important credit derivative instrument in the financial market. The method works exquisitely well even for large dimensions making it a valuable tool for simulating from many different classes of copulas including those which have been difficult to sample from using traditional techniques.     

A model for optimal stopping in advertisement

In this work we propose a model for optimal advertisement in new product diffusion based on the Bass model and assuming that the effect of the environmental pressure in the diffusion of the product is subject to a stochastic dependence. The optimal stopping problem is reduced to a free boundary problem which is analyzed and solved numerically, in order to determine an optimal stopping rule for the advertisement campaign. The numerical solution is obtained through a policy iteration like contraction scheme, the convergence properties of which are studied in detail. Furthermore, the expected time until the optimal stopping of the campaign is estimated. Finally, a combined optimal stopping and control problem for the optimization of the advertisement effectiveness is also proposed and solved numerically. Our results are expected to provide useful guidelines for campaign managers, for the choice of effectiveness and duration of an advertisement campaign.     

A GREEDY ALGORITHM FOR SPARSE PRECISION MATRIX APPROXIMATION

Precision matrix estimation is an important problem in statistical data analysis.This paper proposes a sparse precision matrix estimation approach,based on CLIME estimator and an efficient algorithm GISSρ that was originally proposed for l1 sparse signal recov-ery in compressed sensing.The asymptotic convergence rate for sparse precision matrix estimation is analyzed with respect to the new stopping criteria of the proposed GISSρ algorithm.Finally,numerical comparison of GISSρ with other sparse recovery algorithms,such as ADMM and HTP in three settings of precision matrix estimation is provided and the numerical results show the advantages of the proposed algorithm.     

A two-step sequential procedure for detecting an epidemic change

The typical approach in change-point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, and this is our approach, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. In this paper we focus on epidemic changes, that is, a first change (the outbreak) when there is a change in the distribution, and a second change, when the process regains its ordinary structure. Based on the counting process related to the original process observed at equidistant time points, we propose some stopping rules for this to happen and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes, extreme value asymptotics for Gaussian processes, and the law of the iterated logarithm.     

Optimal Stopping of One-Dimensional Diffusions

In this paper we consider an optimal stopping problem for a time-homogeneous, onedimensional, regular diffusion. An essential tool in our approach is the MARTIN boundary theory. It is possible to determine explicitly the representing measure of a given β-excessive function. It is seen that this correspondence may be used to construct optimal stopping rules. In some specific cases, as demonstrated in the paper, the solution is reached directly and with ease. The so called condition of “smooth pasting” is seen to be a simple consequence of our results.     

AIR Tools — A MATLAB package of algebraic iterative reconstruction methods

We present a MATLAB package with implementations of several algebraic iterative reconstruction methods for discretizations of inverse problems. These so-called row action methods rely on semi-convergence for achieving the necessary regularization of the problem. Two classes of methods are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a new “training” algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP criterion; for the first two methods “training” can be used to find the optimal discrepancy parameter.     

ADD-heuristics' starting procedures for capacitated plant location models

This paper is concerned with heuristics for capacitated plant location models where locations have different capacities. In this case ADD-heuristics normally lead to bad solutions. We present some starting procedures (priority rules) in order to overcome this difficulty. Finally, we report numerical results, including comparisons between ADD-heuristics with starting procedures and DROP-heuristics.     

Local asymptotic normality of a sequential model for marked point processes and its applications

This paper deals with statistical inference problems for a special type of marked point processes based on the realization in random time intervals [0,_u]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times _u satisfying them is proposed. Using these stopping rules, one can treat the processes within the framework of LAN, which yields asymptotic optimalities of various inference procedures. Applications for compound Poisson processes and continuous time Markov branching processes (CMBP) are discussed. Especially, asymptotically uniformly most powerful tests for criticality of CMBP can be obtained. Such tests do not exist in the case of the non-sequential approach. Also, asymptotic normality of the sequential maximum likelihood estimators (MLE) of the Malthusian parameter of CMBP can be derived, although the non-sequential MLE is not asymptotically normal in the supercritical case.