Minimax stopping times for I.I.D. Random variables

Minimax optimal stopping times and minimax worst—case distributions are found for the problem of stopping a sequence of uniformly bounded i.i.d. random variables in a cost and a discount model when only the mean and/or the variance (and not the complete distribution) of the random variables is known     

On the speed of convergence of the distribution of random sums of weighted independent variables

This study provides sufficient conditions under which the uniform distance between the distribution of random sums of weighted random variables and the normal distribution is of order n^-a(log n)^a+b, for α <1/2, n^-1/2(log n)^b+3/2, for b > -3/2, and n^-1/2log₂n for b = -3/2. Extensions to one-sided linear processes are also discussed.     

Online Change Detection for a Poisson Process with a Phase-Type Change-Time Prior Distribution

Abstract

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.     

Bounds for optimal stopping regions in problems with nonlinear costs of observations

In problems of optimal stopping for F((W(t)+x)⁺) -c(s + t) with W(t) denoting Brownian motion, bounds for the optimal stopping regions are derived which describe the growth of the optimal boundaries. This generalizes results by lrle (1988) where the case f(x) = x was treated. Extensions are given when positive part is replaced by absolute value and also when a random walk takes the place of Brownian motion.     

Optimal Stopping for Dynamic Recruitment Problem with Probabilistic Loss of Candidates

Abstract

A problem that often arises in the recruitment process is that the recruitment firms face possible loss of candidates. The loss of candidates can induce a loss cost to firms which need to restart the recruitment process. In this article, we model the recruitment process as a discrete-time stochastic optimal stopping problem with a finite planning horizon, where candidates may be hired by other firms during the period of waiting for employment with a loss probability. An optimal decision rule is presented to maximize the benefit of the recruitment firm. This decision rule demonstrates that the threshold of direct employment will be reduced as the loss probability (or the loss cost) is increasing. In addition, we find that new applicants are hardly being directly employed when the remaining time to the deadline is very long. Finally, a numerical example is given to illustrate the effectiveness of the proposed decision rule.     

On-line quality control procedures for a random walk model with measurement error

An economical on-line quality control procedure is given for cases where measurement error u present.The unobservable quality characteristic procw is assumed to follow a normal random walk.model independent of the measurement error which is white noise. The off-target loss function used is proportional to the squared deviation from the target value.The procw is inspected at regular intervals of m time units and adjusted at the end of a cycle when the process goes out of the control limit d.The limning long-run average cost rate is evaluated.The optimum value of the control parameters,the inspection interval m,and the control limit d are obtained by minimlzlng the long-run average cost rate.When m is small and d is large, explicit expressions for the control parameters are given.     

Asymptotically optimal sequential tests for nonhomogeneous processes

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.     

Optimal design for flexible operation of the post-combustion CO2 capture plant with uncertain economic factors

The ultimate benefit of flexible operation of the post-combustion CO₂ capture (PCC) plant depends on the ability to optimally balance between many competing factors, including the additional capital investment and operating cost savings. In this work, a large number of scenarios are constructed by considering combinations of possible realizations of the uncertain economic factors such as energy cost profile, emission penalty and value of captured CO₂. Then, the design choices like the size of the storage tanks and the regeneration capacity are optimized by minimizing an overall cost averaged over all the scenarios. The optimal design problem is naturally formulated as a two-stage stochastic program. This multi-scenario optimal design is compared with the design that minimizes the overall cost for just a single nominal scenario as well as the design that minimizes the cost averaged over the worst-case scenarios.     

Central limit theorems for nonparametric estimators with real‐time random variables

In this article, asymptotic theories for nonparametric methods are studied when they are applied to real‐time data. In particular, we derive central limit theorems for nonparametric density and regression estimators. For this we formally introduce a sequence of real‐time random variables indexed by a parameter related to fine gridding of time domain (or fine discretization). Our results show that the impact of fine gridding is greater in the density estimation case in the sense that strong dependence due to fine gridding severely affects the major strength of nonparametric density estimator (or its data‐adaptive property). In addition, we discuss some issues about nonparametric regression model with fine gridding of time domain.     

Computational uncertainty quantification for a clarifier-thickener model with several random perturbations: A hybrid stochastic Galerkin approach

Continuous sedimentation processes in a clarifier-thickener unit can be described by a scalar nonlinear conservation law whose flux density function is discontinuous with respect to the spatial position. In the applications of this model, which include mineral processing and wastewater treatment, the rate and composition of the feed flow cannot be given deterministically. Efficient numerical simulation is required to quantify the effect of uncertainty in these control parameters in terms of the response of the clarifier-thickener system. Thus, the problem at hand is one of uncertainty quantification for nonlinear hyperbolic problems with several random perturbations. The presented hybrid stochastic Galerkin method is devised so as to extend the polynomial chaos approximation by multiresolution discretization in the stochastic space. This approach leads to a deterministic hyperbolic system, which is partially decoupled and therefore suitable for efficient parallelisation. Stochastic adaptivity reduces the computational effort. Several numerical experiments are presented.     

Parameter change test for random coefficient integer-valued autoregressive processes with application to polio data analysis

Abstract.  In this paper, we consider the problem of testing for a parameter change in a first-order random coefficient integer-valued autoregressive [RCINAR(1)] model. We employ the cumulative sum (CUSUM) test based on the conditional least-squares and modified quasi-likelihood estimators. It is shown that under regularity conditions, the CUSUM test has the same limiting distribution as the supremum of the squares of independent Brownian bridges. The CUSUM test is then applied to the analysis of the monthly polio counts data set.     

Up-and-Down Designs for Selecting the Dose with Maximum Success Probability

Abstract

Assume that the probability of success is unimodal as a function of dose, such as may be the case when too much of a drug is toxic and too little is ineffective. We characterize a class of up-and-down designs, that is, treatment allocation methodologies, for identifying the dose that maximizes the patients' success probability. These designs are constructed to use accruing information to limit the number of patients that are exposed to doses with high probabilities of failure. This treatment allocation procedure is motivated by Kiefer–Wolfowitz's stochastic approximation procedure. However, we take the response to be binary and the possible treatment space to be a lattice. The procedure is shown to allocate treatments to pairs of subjects in a way that causes the treatment distribution to center around the treatment with maximum success probability. The procedure defines a nonhomogeneous random walk, so well-known theory is used to explicitly characterize the treatment distribution. As an estimator of the best dose, the mode of the empirical treatment distribution is shown to converge faster than does the last dose allocated, which is used as an estimator of the optimal dose in stochastic approximation procedures.