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.     

Sequentially Updated Residuals and Detection of Stationary Errors in Polynomial Regression Models

Abstract

The question whether a time series behaves as a random walk or as a stationary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This article studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a Monitoring procedure given by a control chart; i.e., a stopping time, which is related to a well-known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process that implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study.     

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     

A Detailed Model for the Sintering of Polydispersed Nanoparticle Agglomerates

In this study the coagulation, condensation, and sintering of nanoparticles is investigated using a stochastic particle model. Each stochastic particle consists of interacting polydisperse primary particles that are connected to each other. In the model sintering occurs between each individual pair of neighboring primary particles. This is important for particles in which the range of the size of the primary particles varies significantly. The sintering time is obtained from the viscous flow model. The model is solved using a stochastic particle algorithm. The particles are represented in a binary tree that contains the connectivity as well as the degree of sintering information. Particles are forme, coagulate, sinter, and experience condensation according to known rate laws. The particle binary tree, along with it the degree of sintering, is updated after each time step according to the rates of the different processes. The stochastic particle method uses the technique of fictitious jumps and linear process deferment. The theoretical results are fitted against experimental values for the formation of SiO ₂ nanoparticles and computer generated TEM pictures are presented and compared to experiments.     

Asymptotically Pointwise Optimal Change Detection in Multiple Channels

Abstract

A Bayesian multichannel change-point detection problem is studied in the following general setting. A multidimensional stochastic process is observed; some or all of its components may experience changes in distribution, simultaneously or not. The loss function penalizes for false alarms and detection delays, and the penalty increases with each missed change-point. For wide classes of stochastic processes, with or without nuisance parameters and practically any joint prior distribution of change-points, asymptotically pointwise optimal (APO) rules are obtained, translating the classical concept of Bickel and Yahav to the sequential change-point detection. These APO rules are attractive because of their simple analytic form and straightforward computation. An application to a multidimensional autoregressive time series is shown.     

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.     

SPURIOUS REGRESSIONS BETWEEN I(d) PROCESSES

Abstract. This paper develops an analytical study for the nonsense or spurious regressions that are generated by quite general integrated (of order d) random processes. In doing this, we generalize the work of Phillips (Understanding spurious regressions in econometrics. J. Econ. 33 (1986), 311–40) who provided an analytical study of linear regressions involving only I(1) stochastic processes. Our generalization of Phillips' work to the I(d) case is made employing fractional differencing techniques.     

Complete convergence of stochastic approximation algorithm in ℝd under random noises

In this article, we study a stochastic approximation algorithm that approximates the exact root θ of a function M defined in ?^d into ?^d. The function M cannot be known exactly, but only noisy measurements are available at each point x_n with the error ξ_n. The sequence of noise (_ξn)n is random; we treat both cases where it is independent and dependent and we establish the complete convergence of the approximated sequence of θ.     

Power of the MOSUM test for online detection of a transient change in mean

Abstract

In this article we discuss an online moving sum (MOSUM) test for detection of a transient change in the mean of a sequence of independent and identically distributed (i.i.d.) normal random variables. By using a well-developed theory for continuous time Gaussian processes and subsequently correcting the results for discrete time, we provide accurate approximations for the average run length (ARL) and power of the test. We check theoretical results against simulations, compare the power of the MOSUM test with that of the cumulative sum (CUSUM), and briefly consider the cases of nonnormal random variables and weighted sums.     

Pareto Optimality in a Bicriterion Optimal Stopping Problem

Abstract

This article concerns a characterization of a vector-valued optimal stopping problem with only two reward functions. Under certain conditions, we give an explicit characterization of the Pareto stopping times for the present problem via a scalar-valued double optimal stopping problem. The latter problem is a natural extension of the classical McDonald-Siegel optimal stopping problem with one stopping rule.     

Design and assessment of delay timer alarm systems for nonlinear chemical processes

In process and manufacturing industries, alarm systems play a critical role in ensuring safe and efficient operations. The objective of a standard industrial alarm system is to detect undesirable deviations in process variables as soon as they occur. Fault detection and diagnosis systems often need to be alerted by an industrial alarm system; however, poorly designed alarms often lead to alarm flooding and other undesirable events. In this article, we consider the problem of industrial alarm design for processes represented by stochastic nonlinear time‐series models. The alarm design for such complex processes faces three important challenges: (1) industrial processes exhibit highly nonlinear behavior; (2) state variables are not precisely known (modeling error); and (3) process signals are not necessarily Gaussian, stationary or uncorrelated. In this article, a procedure for designing a delay timer alarm configuration is proposed for the process states. The proposed design is based on minimization of the rate of false and missed alarm rates—two common performance measures for alarm systems. To ensure the alarm design is robust to any non‐stationary process behavior, an expected‐case and a worst‐case alarm designs are proposed. Finally, the efficacy of the proposed alarm design is illustrated on a non‐stationary chemical reactor problem. © 2017 American Institute of Chemical Engineers AIChE J, 63: 77–90, 2018     

Bayesian bandits in clinical trials

Suppose two treatments with binary responses are available for patients with some disease. Sequential allocation rules based on the theory of Bayesian bandit processes are examined. The patient horizon is assumed to be random and the objective is to maximize the total expected number of successes. This problem is equivalent to the classical two—armed bandit problem if we assume that the only information acquired during the trial about the patient horizon is whether or not it has so far been reached. When one treatment has a known success rate, the optimal allocation rule may sometimes be expressed in terms of dynamic allocation indices. The calculation of the indices is discussed, and in particular error bounds for the accuracy of the calculation are given in terms of the starting point of the backward induction process. When both treatments have unknown success rates the use of dynamic allocation indices with geometric discounting is suggested. Simulation results indicate that this rule works well even when the distribution of the number of the patients is not geometrical, and that the choice of the discount factor is also not crucial.     

Integration of CARMA processes and spot volatility modelling

Continuous‐time autoregressive moving average (CARMA) processes with a non‐negative kernel and driven by a non‐decreasing Lévy process constitute a useful and very general class of stationary, non‐negative continuous‐time processes which have been used, in particular for the modelling of stochastic volatility. In the celebrated stochastic volatility model of Barndorff‐Nielsen and Shephard (2001) , the spot (or instantaneous) volatility at time t, V(t), is represented by a stationary Lévy‐driven Ornstein‐Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, limiting its ability to generate integrated volatility sequences, , with autocorrelation functions resembling those of observed realized volatility sequences. (A realized volatility sequence is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically 1 day.) If instead of the stationary Ornstein–Uhlenbeck process, we use a CARMA process to represent spot volatility, we can overcome the restriction to exponentially decaying autocorrelation function and obtain a more realistic model for the dependence observed in realized volatility. In this article, we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to a daily realized volatility sequence for the Deutsche Mark/ US dollar exchange rate.     

Exponential inequalities for Mann’s stochastic algorithm

Abstract

In this article, we investigate the problem of approximating the fixed point for some function using a Mann iterative process with random errors. After establishing some exponential inequalities, we prove the complete convergence of Mann’s algorithm toward the fixed point and deduce a confidence interval for this one. In addition, we establish the convergence rate of Mann’s algorithm. Several numerical examples are sketched to illustrate the performance of the proposed algorithm.     

Contributions to the Stochastic Theory of Chromatographic Kinetics

Abstract

The stochastic theory of chromatographic kinetics is extended in two directions: (a) To include diffusion effects, and (b) to treat the n-site adsorption problem. In the first case we show that the solution to a first passage problem yields the moments of residence time in the mobile phase in a rather simple form. For the second problem we show that the central limit theorem allows us to deduce rather general results about the distribution of residence time.     

The maximum of a sequence with prior information

Let X₁...,X_n denote a random sample of size n form distribution function F. The X's are observed one at a time in sequence; the problem is to stop at a maximum of the n observations, where recall is not permitted and where the loss is 0 if a maximum is selected and 1 if not. Here it is assumed that the distribution function is from a Dirichlet process with known parameter. Sequential stopping rules are developed and the probabilities of maximal selection computed. robustness of the procedure is investigated in an example.     

Optimal procurement contract selection with price optimization under uncertainty for process networks

In this work, we propose extending the production planning decisions of a chemical process network to include optimal contract selection under uncertainty with suppliers and product selling price optimization. We use three quantity-based contract models: discount after a certain purchased amount, bulk discount, and fixed duration contracts. We propose the use of general regression models to describe the relationship between selling price, demand, and possibly other predictors, such as economic indicators. For illustration purposes, we consider three demand-response models (i.e., selling price as a function of demand) that are typically encountered in the literature: linear, constant-elasticity, and logit. We develop a mixed-integer nonlinear two-stage stochastic programming that accounts for uncertainty in both supply (e.g., raw material spot market price) and demand (random nature of the residuals of the regression models) for the planning of the process network. The proposed method is illustrated with two numerical examples of chemical process networks.     

Some Almost-Sure Convergence Properties Useful in Sequential Analysis

Abstract

Kim and Nelson propose sequential procedures for selecting the simulated system with the largest steady-state mean from a set of alternatives that yield stationary output processes. Each procedure uses a triangular continuation region so that sampling stops when the relevant test statistic first reaches the region's boundary. In applying the generalized continuous mapping theorem to prove the asymptotic validity of these procedures as the indifference-zone parameter tends to zero, we are given (i) a sequence of functions on the unit interval (which are right-continuous with left-hand limits) converging to a realization of a certain Brownian motion process with drift; and (ii) a sequence of triangular continuation regions corresponding to the functions in sequence (i) and converging to the triangular continuation region for the Brownian motion process. From each function in sequence (i) and its corresponding continuation region in sequence (ii), we obtain the associated boundary-hitting point; and we prove that the resulting sequence of such points converges almost surely to the boundary-hitting point for the Brownian motion process. We also discuss the application of this result to a statistical process-control scheme for autocorrelated data and to other selection procedures for steady-state simulation experiments.

     

Stochastic Ordering in the Qualitative Assessment of the Performance of Simultaneous Schemes for Bivariate Processes

Abstract

Misleading signals (MS) are likely to happen while using a simultaneous scheme to control the mean vector (μ) and the covariance matrix (Σ) of a bivariate process. They correspond to valid signals that lead to a misinterpretation of a shift in μ (resp. Σ) as a shift in Σ (resp. μ). Following previous work, focused on the quantitative assessment of the probabilities of misleading signals (PMS) in simultaneous schemes for bivariate processes, we now make use of stochastic ordering to qualitatively assess the impact of changes in μ and Σ in those probabilities.     

Editor's Special Invited Paper: Second-Guessing Clinical Trial Designs

Abstract

Modern change-point detection had its origins about 50 years ago in the work of Page, Shiryaev, and Lorden, who focused on sequential detection of a change-point in a sequence of observations. Motivation often arose from sequential quality control: to detect a disruption in the quality of a continuous production process. More recently, motivation from a broad range of applications has led to a variety of different problem formulations. In this article I will review this history with particular attention to a selected subset of applications arising in biology and to common features of different likelihood-based formulations.