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Modern computing and communication technologies can make data collection procedures very efficient, while our ability to analyze and extract information from large data sets is hard-pressed to keep up with our capacity for data collection. If we are interested in learning a classification/prediction rule that was not considered in the original data collection procedure, then there is usually a lack of label information in the original data. Thus, how to start with minimum number of labeled data and aggressively selected most informative samples for being labeled becomes an important issue. The thoughts of active learning, with subjects to be selected sequentially without using label information, is a possible outlet for this situation. In addition, if we can identify variables, from the lengthy variable list of the data set, for constructing an effective classification rule with good interpretation ability will be better. Here, we propose an active learning procedure targeted the maximum area under the receiver operating characteristic curve via a commonly use logistic model, which can sequentially select the effective informative subjects and variables, simultaneously. We discuss the asymptotic properties of the proposed procedure and illustrate it with some synthesized and real data sets. [ABSTRACT FROM AUTHOR]

Sequential change-point analysis, which identifies a change of probability distribution in an infinite sequence of random observations, has important applications in many fields. A good method should detect a change point as soon as possible, and keep a low amount of false alarms. As one of the most popular methods, Shiryaev-Roberts (SR) procedure holds many optimalities. However, its implementation requires the pre-change and post-change distributions to be known, which is not achievable in practice. In this paper, we construct a nonparametric version of the SR procedure by embedding different versions of empirical likelihood, assuming two training samples, before and after change, are available for parameter estimations. Simulations are conducted to compare the performance of the proposed method with existing methods. The results show that when the underlying distribution is unknown, and training sample sizes are small, the proposed modified procedure shows advantage by giving a smaller delay of detection. [ABSTRACT FROM AUTHOR]

We give a survey of the results related to extensions of the Kolmogorov inequality for the distribution of the absolute value of the maximum of the sum of centered independent random variables to the case of martingales considered at random stopping times. [ABSTRACT FROM AUTHOR]

What is the number of rolls of fair six-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10 − 4 . This is a solution to a puzzle suggested by DasGupta in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic finite‐time stability (SFTS) criteria for switched stochastic nonlinear systems (SSNS) with nonlocal Lipschitz coefficients are established. The main tools that we used here are mode‐dependent limiting average dwell time (MD‐LADT), stopping time technology and the stochastic analysis theory. The established criteria show that the diffusion coefficient, the drift coefficient and the switching law play crucial roles in obtaining the SFTS for the SSNS. Besides, our criteria provide some approaches on how to design the switching law to achieve the SFTS if a switched stochastic system contains SFTS subsystems and stochastic finite‐time instable (SFTI) subsystems simultaneously. Finally, an example is given to illustrate the importance and usefulness of the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the statistical concept of causality in continuous time within filtered probability spaces based on Granger's definition of causality. The given causality concept is connected to the optional and predictable processes important in stochastic integration. More precisely, we establish that the preservation of predictability with respect to larger filtrations is implied by the considered notion of (self-)causality. We also consider the connections between the given causality concept and the optional and predictable projections of a stochastic process, which play an important role in the general theory of stochastic processes, semimartingale theory, and stochastic calculus. Some results show that the (self-)causality implies indistinguishability of the optional (or predictable) projections with respect to considered filtrations from those with respect to larger filtrations. [ABSTRACT FROM AUTHOR]

Sequential change point tests aim at giving an alarm as soon as possible after a structural break has occurred while controlling the asymptotic false alarm rate. For such tests it is of particular importance to understand how quickly a break is detected. In this paper, we derive the asymptotic distribution of the delay times for sequential change point procedures based on U-statistics. This includes the difference-of-means (DOM) sequential test that has been discussed previously but also a new robust Wilcoxon sequential change point test. Based on these results we can compare the detection delay of both procedures instead of having to rely on simulations only. Indeed, the Wilcoxon sequential procedure has a smaller detection delay for heavier tailed distributions which is also confirmed by simulations. While in the previous literature for the DOM test only results for early change points were obtained, we derive the asymptotic distribution of the delay times for both early as well as late change points. Finally, we evaluate how well the asymptotic distribution approximates the actual stopping times for finite samples via a simulation study. • Asymptotic distribution of delay times for sequential tests of U -statistics. • This includes a robust Wilcoxon sequential procedure. • This includes late (linear and superlinear) changes. • For early changes the approximation improves upon existing approximations. [ABSTRACT FROM AUTHOR]

This paper studies the issue of coordinating equipment maintenance operations with capital investment strategy in the presence of random equipment failures. The traditional approach, developed by Kamien and Schwartz (KS) in their celebrated paper published in 1971, is to formulate the problem as a deterministic optimal control problem with the probability of machine failure as the state variable. With this approach, the optimal policy is deterministic. As a major departure from the KS approach, we explicitly model the underlying stochastic process of machine failures. Our analysis of the stochastic dynamic programming model offers new insights into the problem. Under a long planning horizon with a limited replacement opportunity, each individual machine serves as a revenue generator and contributes a significant amount to the profit of the system. In contrast, when the replacement budget is quite generous over a relatively short planning horizon, adding one extra machine only helps as a backup for unexpected failures of the machines purchased before it. An interesting result derived from this comparison is that a deterministic policy turns out to be optimal for the former, while a state-contingent policy must be applied to the latter. In other words, the deterministic KS approach does not work in general when a chain of machine replacement is considered. We further characterize the effects of the discount rate, productivity deterioration, learning, decision delay, and technology advancement on the optimal policy. [ABSTRACT FROM AUTHOR]

We study a class of non-linear Backward stochastic differential equations (BSDE) with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition ξ is allowed to take the value +∞, i.e., singular. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value ∞ at terminal time S. Our goal is to show existence of solutions to the BSDE for a range of singular terminal values under the assumption that S is solvable. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., it is continuous at time S and attains the terminal value with probability 1. We consider three types of terminal values: 1) Markovian: i.e., ξ is of the form ξ = g(ΞS) where Ξ is a continuous Markovian diffusion process, S is a hitting time of Ξ and g is a deterministic function 2) terminal conditions of the form ξ1 = ∞· 1(τ≥) and 3) ξ2 = ∞· 1(τ≥) where τ is another stopping time. For general ξ we prove that minimal supersolution has a limit at time S provided that F is left continuous at time S. Finally, we discuss the implications of our results about Markovian terminal conditions to the solution of non-linear elliptic PDE with singular boundary conditions. [ABSTRACT FROM AUTHOR]

In this study, we consider an optimal control problem driven by a stochastic differential system with a stopping time terminal cost functional. To solve the optimal control problem under the stopping time terminal cost functional, we introduce a multi-time state optimal control systems, and prove that the multi-time state optimal control systems is a near-optimal control problem. We use the near-optimal control problem to take place the original optimal control problem under the cost functional with a stopping time terminal. Then, we establish the stochastic maximum principle for this kind of optimal control problem by introducing a discrete terminal system. Finally, we provide an example to describe the main results of this study. [ABSTRACT FROM AUTHOR]

All the available modes of travel and their respective travel parameters must be known to the commuters before their trip. Otherwise they may either spend more money or more time for the trip. In addition to this, recent pandemic, rapidly spreading novel corona virus is demanding a smart solution for contactless commuting. This paper suggests a practical solution to make both the above possible and it emphasizes the applicability of two developed android applications, one for travel data collection and another to predict travel time for a multimodal trip within the study area. If the whole trip is by a single mode, the user can get the corresponding travel time estimate from "Google maps". But, if the trip is by multiple modes, it is not possible to get the total travel time estimate for the whole trip at a time from "Google maps". A separate travel mode for "auto" is unavailable in "Google maps" alongside drive, two-wheeler, train or bus and walk alternatives. It is also observed that the travel time estimate of "Google maps" for the city buses is inaccurate. Hence, the two modes (Buses and Autos) were chosen for the study. Unless and until the travel times and stopping times of the two modes are known, it is not possible to predict their trip times. Hence, the mobility analysis was performed for the two modes in the study area to find their respective average travel rate at peak hours, across 15 corridors and the results were presented. [ABSTRACT FROM AUTHOR]

Cognitive radio (CR) is a revolutionary paradigm to migrate the spectrum scarcity problem in wireless networks. In view of the spectrum scarcity of unmanned aerial vehicles (UAVs) communication system, cooperative spectrum sensing (CSS) has emerged as a key function of CR to identify the available spectrum for UAV sensing nodes. However, the flexible locations of flying UAVs make CSS a difficult task, which in turn makes it difficult for CSS performance to meet UAV requirements. In this paper, we design an intra-frame cooperation way to achieve CSS for cognitive UAV networks (CUAVNs), which provides the same CSS performance as the traditional inter-frame cooperation and higher throughput for UAVs. Furthermore, considering certain performance requirements in CUAVNs, we propose sequential probability ratio test (SPRT)-based CSS method, and further make an in-depth analysis of relationship among performance indices of CSS and approximate the actual detection performance and the stopping time by ignoring the excess over the boundaries and the random walk. Finally, numerical results corroborate the effectiveness and correctness of our theoretical analysis, while comparison results also represent the superiority of our proposal. [ABSTRACT FROM AUTHOR]

We consider the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution. This problem is investigated in a sequential setup under mild assumptions on the underlying random process. The optimal method minimizes the expected number of samples while ensuring that the average detection/estimation errors do not exceed a certain level. After converting the constrained problem to an unconstrained one, we characterize the general solution by a nonlinear Bellman equation, which is parameterized by a set of cost coefficients. A strong connection between the derivatives of the cost function with respect to the coefficients and the detection/estimation errors of the sequential procedure is derived. Based on this fundamental property, we further show that for suitably chosen cost coefficients the solutions of the constrained and the unconstrained problem coincide. We present two approaches to finding the optimal coefficients. For the first approach, the final optimization problem is converted into a linear program, whereas the second approach solves it with a projected gradient ascent. To illustrate the theoretical results, we consider two problems for which the optimal schemes are designed numerically. Using Monte Carlo simulations, it is validated that the numerical results agree with the theory. [ABSTRACT FROM AUTHOR]

In view of the spectrum scarcity of unmanned aerial vehicles' (UAVs) communication systems, cooperative spectrum sensing (CSS) is used to identify the available spectrum for cognitive UAV networks (CUAVNs). Due to the flexible locations of flying UAVs, we first design an intraframe cooperation way to replace the traditional interframe cooperation to achieve CSS in this article. Considering certain performance requirements in CUAVNs, we perform an in-depth analysis of the relationship among performance indices of sequential probability ratio test (SPRT)-based CSS and approximate the actual detection performance and the stopping time by ignoring the excess over the boundaries and the random walk. Finally, numerical results corroborate the effectiveness and correctness of our theoretical analysis, and comparison results represent the superiority of our proposal. [ABSTRACT FROM AUTHOR]

The paper considers the problem of optimal control in models with disorder and provides the computational scheme of its solution. The proposed method is based on replacing the disorder with its estimate by the stopping time. The calculations take place on the binary tree with nodes divided into two classes. The division of tree nodes into two classes is carried out using a decision rule associated either with testing two hypothesis, or with the optimal stopping of a Markov process developing on a binary tree. The estimate of the disorder is the moment at which the walk on the tree is placed in a node of the first class for the first time. The example of the optimal control problem is related to the problem of quantile hedging of a payoff. [ABSTRACT FROM AUTHOR]

Bias approximation has played an important rôle in statistical inference, and numerous bias calculation techniques have been proposed under different contexts. We provide a unified approach to approximating the bias of the maximum likelihood estimator and the l2 penalized likelihood estimator for both linear and nonlinear models, where the design variables are allowed to be random and the sample size can be a stopping time. The proposed method is based on the Woodroofe–Stein identity and is justified by very weak approximations. The accuracy of the derived bias formulas is assessed by simulation for several examples. The bias of the ridge estimator in high‐dimensional settings is also discussed. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic properties of the stopping times based on the monitoring tests for a bubble, that is, ADF‐type and CUSUM‐type tests. We derive the stochastic orders of these stopping times and find that they depend on the relative position of the emerging date of a bubble. We show that when a bubble emerges early in the monitoring period, the diverging rate of the stopping time based on the CUSUM‐type detector is slower than that based on the ADF‐type detector, which implies that the former tends to find a bubble earlier than the latter, whereas this advantage disappears for the late emerging bubble. We also derive the limiting distribution of the stopping time based on the CUSUM detector. The finite sample simulations support the theoretical result. [ABSTRACT FROM AUTHOR]

Braking distances play an important role in the organization of rail traffic and exploitation of rail vehicles. The braking distances of rail vehicles affect several factors that cannot be defined and determined precisely. In this paper, a mathematical model, for train moving through braking, is presented by taking into account the relevant train braking resistance and a procedure is given for solving a differential equation of movement of the rail vehicle during braking. The procedure allows determining influential factors that have an effect on the length of braking distances. The influence and the changes of specific resistance during the movement of the train, braking force, and adhesion between wheel and rail during the braking process are presented. The results obtained with the presented mathematical model are accurate and match with the results obtained experimentally. [ABSTRACT FROM AUTHOR]

In this paper, we present a switching property for squared Bessel process. We prove that the starting point and the time parameter can be in a sense switched. As a consequence, we obtain new distributional dependencies among a squared Bessel process R, a Brownian motion with drift B μ , μ ≥ 0 , B t μ ≔ B t + μt for a Brownian motion B , and an integral functional of geometric Brownian motion A(μ) connected by the Lamperti relation. Finally, we deduce a new explicit form of the joint distribution of the Lamperti triple R T A t μ exp 2 B t μ for fixed t > 0 andT > 0. [ABSTRACT FROM AUTHOR]

Two optimal stopping problems for geometric random walks with the observer's power payoff function, on the finite and infinite horizons, are solved. For these problems, an explicit form of the cut value and also optimal stopping rules are established. It is proved that the optimal stopping rules are nonrandomized thresholds and describe the corresponding free boundary. An explicit form of the free boundary is presented. [ABSTRACT FROM AUTHOR]

Background: The Elliot wave principle commonly characterizes the impulsive and corrective wave trends for both financial market trends and electrocardiograms. The impulsive wave trends of electrocardiograms can annotate several wave components of heart-beats including pathological heartbeat waveforms. The stopping time inquires which ordinal element satisfies the assumed mathematical condition within a numerical set. The proposed work constitutes several algorithmic states in reinforcement learning from the stopping time decision, which determines the impulsive wave trends. Each proposed algorithmic state is applicable to any relevant algorithmic state in reinforcement learning with fully numerical explanations. Because commercial electrocardiographs still misinterpret myocardial infarctions from extraordinary electrocardiograms, a novel algorithm needs to be developed to evaluate myocardial infarctions. Moreover, differential diagnosis for right ventricle infarction is required to contraindicate a medication such as nitroglycerin.Methods: The proposed work implements the stopping time theory to impulsive wave trend distribution. The searching process of the stopping time theory is equivalent to the actions toward algorithmic states in reinforcement learning. The state value from each algorithmic state represents the numerically deterministic annotated results from the impulsive wave trend distribution. The shape of the impulsive waveform is evaluated from the interoperable algorithmic states via least-first-power approximation and approximate entropy. The annotated electrocardiograms from the impulsive wave trend distribution utilize a structure of neural networks to approximate the isoelectric baseline amplitude value of the electrocardiograms, and detect the conditions of myocardial infarction. The annotated results from the impulsive wave trend distribution consist of another reinforcement learning environment for the evaluation of impulsive waveform direction.Results: The accuracy to discern myocardial infarction was found to be 99.2754% for the data from the comma-separated value format files, and 99.3579% for those containing representative beats. The clinical dataset included 276 electrocardiograms from the comma-separated value files and 623 representative beats.Conclusions: Our study aims to support clinical interpretation on 12-channel electrocardiograms. The proposed work is suitable for a differential diagnosis under infarction in the right ventricle to avoid contraindicated medication during emergency. An impulsive waveform that is affected by myocardial infarction or the electrical direction of electrocardiography is represented as an inverse waveform. [ABSTRACT FROM AUTHOR]

Modeling correlated or highly stratified multiple‐response data is a common data analysis task in many applications, such as those in large epidemiological studies or multisite cohort studies. The generalized estimating equations method is a popular statistical method used to analyze these kinds of data, because it can manage many types of unmeasured dependence among outcomes. Collecting large amounts of highly stratified or correlated response data is time‐consuming; thus, the use of a more aggressive sampling strategy that can accelerate this process—such as the active‐learning methods found in the machine‐learning literature—will always be beneficial. In this study, we integrate adaptive sampling and variable selection features into a sequential procedure for modeling correlated response data. Besides reporting the statistical properties of the proposed procedure, we also use both synthesized and real data sets to demonstrate the usefulness of our method. [ABSTRACT FROM AUTHOR]

We consider sequential detection based on quantized data in the presence of eavesdropper. Stochastic encryption is employed as a counter measure that flips the quantization bits at each sensor according to certain probabilities, and the flipping probabilities are only known to the legitimate fusion center (LFC) but not the eavesdropping fusion center (EFC). As a result, the LFC employs the optimal sequential probability ratio test (SPRT) for sequential detection whereas the EFC employs a mismatched SPRT (MSPRT). We characterize the asymptotic performance of the MSPRT in terms of the expected sample size as a function of the vanishing error probabilities. We show that when the detection error probabilities are set to be the same at the LFC and EFC, every symmetric stochastic encryption is ineffective in the sense that it leads to the same expected sample size at the LFC and EFC. Next, in the asymptotic regime of small detection error probabilities, we show that every stochastic encryption degrades the performance of the quantized sequential detection at the LFC by increasing the expected sample size, and the expected sample size required at the EFC is no fewer than that is required at the LFC. Then the optimal stochastic encryption is investigated in the sense of maximizing the difference between the expected sample sizes required at the EFC and LFC. Although this optimization problem is nonconvex, we show that if the acceptable tolerance of the increase in the expected sample size at the LFC induced by the stochastic encryption is small enough, then the globally optimal stochastic encryption can be analytically obtained; and moreover, the optimal scheme only flips one type of quantized bits (i.e., 1 or 0) and keeps the other type unchanged. [ABSTRACT FROM AUTHOR]

We study a finite horizon optimal multi-modes switching problem with many nodes. The switching is based on the optimal expected profit and cost yields, moreover both sides of the balance sheet are considered. The profit and cost yields per unit time are respectively assumed to be coupled through a coupling term which is the average of profit and cost yields. The corresponding system of Snell envelopes is highly complex, so we consider the aggregated yields where a mean-field approximation is used for the coupling term. First, the problem is formulated by the mean of the Snell envelope of processes. Then, in terms of backward SDEs, the problem is equivalent to a system of mean-field reflected backward SDEs with interconnected and nonlinear obstacles. More precisely, the driver function depends also on the mean of the unknown process (expected profit or cost yields) which makes the mean-field interaction in the driver nonlinear. The first main result of this paper, is to show the existence of a continuous minimal solution of the system of mean-field reflected backward SDEs, which is done by using the Picard iteration method. The second main result concerns the optimality of the switching strategies which we fully characterize. [ABSTRACT FROM AUTHOR]

In this work we discuss almost sure convergence for sums of arbitrarily dependent stochastic sequence under different conditions of Chung's type. Our approach is based on the stopping time technique and the theorem of convergence for martingale difference sequence. Meanwhile, the results here include some relevant classical conclusions. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying stopping time problem of interval-valued differential equation. Based on length constraint, we discuss the stopping time without Lipschitz condition, monotonicity and nonnegativity hypotheses. Then, stopping times with respects to upper and lower boundednesses are studied without monotonicity hypothesis, which is necessary in [27]. The results can be applied to discuss stopping time problem of more general interval-valued differential equations. We also provide some examples to verify our results. [ABSTRACT FROM AUTHOR]

We propose a two-stage sequential method for obtaining tandem-width confidence intervals for a Bernoulli proportion p. The term "tandem-width" refers to the fact that the half-width of the confidence interval is not fixed beforehand; it is instead required to satisfy two different half-width upper bounds, h0 and h1, depending on the (unknown) values of p. To tackle this problem, we first propose a simple but useful sequential method for obtaining fixed-width confidence intervals for p, whose stopping rule is based on the minimax estimator of p. We observe Bernoulli(p) trials sequentially, and for some fixed half-width h = h0 or h1, we develop a stopping time T such that the resulting confidence interval for p, [ ], covers the parameter with confidence at least where is the maximum likelihood estimator of p at time T. Furthermore, we derive theoretical properties of our proposed fixed-width and tandem-width methods and compare their performances with existing alternative sequential schemes. The proposed minimax-based fixed-width method performs similarly to alternative fixed-width methods, while being easier to implement in practice. In addition, the proposed tandem-width method produces effective savings in sample size compared to the fixed-width counterpart and provides excellent results for scientists to use when no prior knowledge of p is available. [ABSTRACT FROM AUTHOR]

Let and denote Markov processes on being the results of the following transformations of a symmetric -stable process on : the first one coincides with the process killed at the stopping time and the second one is determined by its transition probability density defined for , , and by the equality , where g is transition probability density of the process . If those transformations lead us to the same process. In the case of they are different. Some properties of these processes are investigated. We restrict ourselves, in this paper, to the case of . [ABSTRACT FROM AUTHOR]

Joint detection and estimation refers to deciding between two or more hypotheses and, depending on the test outcome, simultaneously estimating the unknown parameters of the underlying distribution. This problem is investigated in a sequential framework under mild assumptions on the underlying random process. We formulate an unconstrained sequential decision problem, whose cost function is the weighted sum of the expected run-length and the detection/estimation errors. Then, a strong connection between the derivatives of the cost function with respect to the weights, which can be interpreted as Lagrange multipliers, and the detection/estimation errors of the underlying scheme is shown. This property is used to characterize the solution of a closely related sequential decision problem, whose objective function is the expected run-length under constraints on the average detection/estimation errors. We show that the solution of the constrained problem coincides with the solution of the unconstrained problem with suitably chosen weights. These weights are characterized as the solution of a linear program, which can be solved using efficient off-the-shelf solvers. The theoretical results are illustrated with two example problems, for which optimal sequential schemes are designed numerically and whose performance is validated via Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

The invariance formulas for stopping times of squared Bessel processes with positive index are presented. For a stopping time τ satisfying some relatively mild assumptions and every positive Borel function f, we have the invariance formula where and γμ is a gamma random variable independent of σ(τ, R(τ)). The second invariance formula is where K is a constant and Δμ is an explicitly given function. The applications of invariance formulas are presented. In particular, new results on the distribution of a killed squared Bessel process, a conditional inverse local time, and the first hitting time of a graph by a squared Bessel process are delivered. [ABSTRACT FROM AUTHOR]

In sequential analysis, an experimenter may gather information regarding an unknown parameter by observing random samples in successive steps. We emphasize a number of specific parametric models under a variety of loss functions. The total number of observations collected at termination is a positive integer-valued random variable, customarily denoted by . The exact probability distribution of N is often hard to obtain. However, under a set of regulatory conditions, our standardized version of the stopping variable from Definition 2.1 in Section 2.2 would follow a normal distribution in the asymptotic sense. In this article, we first show how these regulatory conditions build upon one another in order to conclude the asymptotic normality of such standardized stopping variable, . We provide exploratory data analysis (EDA) via a number of interesting illustrations obtained through large-scale simulation studies. We demonstrate the broad applicability of purely sequential methodologies included in this article along with the appropriateness of our conclusions regarding the asymptotic normality of the standardized stopping variable as a practical guideline. [ABSTRACT FROM AUTHOR]

This paper analyzes the asymptotic performances of fully distributed sequential hypothesis testing procedures as the type-I and type-II error rates approach zero, in the context of a sensor network without a fusion center. In particular, the sensor network is defined by an undirected graph, where each sensor can observe samples over time, access the information from the adjacent sensors, and perform the sequential test based on its own decision statistic. Different from most literature, the sampling process and the information exchange process in our framework take place simultaneously (or, at least in comparable time-scales), thus cannot be decoupled from one another. Our goal is to achieve second-order asymptotically optimal performance at every sensor, i.e., the average detection delay is within a constant gap from the centralized optimal sequential test as the error rates approach zero for the fixed number of sensors. To that end, a type of test procedure that resembles the well-known sequential probability ratio test (SPRT), termed as distributed SPRT (DSPRT) in this paper, is studied based on two message-passing schemes, respectively. The first scheme features the dissemination of the raw samples. In specific, every sample propagates over the network by being relayed from one sensor to another until it reaches all the sensors in the network. Although the sample propagation-based DSPRT is shown to yield the asymptotically optimal performance at each sensor, it incurs excessive inter-sensor communication overhead due to the exchange of raw samples with index information. The second scheme adopts the consensus algorithm, where the local decision statistic is exchanged between sensors instead of the raw samples, thus significantly lowering the communication requirement compared with the first scheme. In particular, the decision statistic for DSPRT at each sensor is updated by the weighted average of the decision statistics in the neighborhood at every message-passing step. We show that, under certain regularity conditions, the consensus algorithm-based DSPRT also yields the second-order asymptotically optimal performance at all sensors given a fixed number of sensors. Our asymptotic analyses of the two message-passing-based DSPRTs are then corroborated by simulations using the Gaussian and Laplacian samples. [ABSTRACT FROM AUTHOR]

In this paper, we obtain fixed-width confidence interval for covariate-adjusted response-adaptive designs. Specifically, we consider logistic regression model and the normal regression model for binary and continuous responses, respectively, both in the situations for presence and absence of treatment–covariate interactions. Simulation study and real-data analysis are carried out. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of sequentially detecting statistical changes. In particular, the focus is on transient change detection, in which a probability minimizing optimal criterion is desirable. This is in contrast with the traditional minimization of the detection delay, proposed in quickest change detection problems.Afinite moving average stopping time is proposed for the general transient change detection problem. The statistical performance of this stopping time is investigated and compared to other methods available in the literature. The proposed stopping time and theoretical findings are applied to quality monitoring, including reliabilitymonitoring in industrial processes and signal quality monitoring in global navigation satellite systems. Numerical simulations are presented to assess the goodness of the presented theoretical results, and the performance of the considered stopping times. This will show the superiority of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper is devoted to establish an existence and uniqueness result of one-dimensional reflectedbackward stochastic differential equations with time-delayed generators (RBSDEswith time-delayed generators, in short). Our proof is based on approximation via a penalization method. [ABSTRACT FROM AUTHOR]

Sequential procedure is developed for minimum risk point estimation of the Lomax distribution under quadratic plus linear cost of sampling of the shape parameter. Secondorder approximations are derived for the expected sample size and regret associated with sequential procedure. [ABSTRACT FROM AUTHOR]

We start proceeding with the stopping time theory in discrete time with the help of the Mizar system [1], [4]. We prove, that the expression for two stopping times k1 and k2 not always implies a stopping time (k1 + k2) (see Theorem 6 in this paper). If you want to get a stopping time, you have to cut the function e.g. (k1 + k2) ∪ T (see [2, p. 283 Remark 6.14]). Next we introduce the stopping time in continuous time. We are focused on the intervals [0, r] where r ∈ R. We prove, that for I = [0, r] or I = [0,+∞[ the set {A ⋂ I : A ∈ Borel-Sets} is a σ-algebra of I (see Definition 6 in this paper, and more general given in [3, p.12 1.8e]). The interval I can be considered as a timeline from now to some point in the future. This set is necessary to define our next lemma. We prove the existence of the σ-algebra of the τ -past, where τ is a stopping time (see Definition 11 in this paper and [6, p.187, Definition 9.19]). If τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we can prove, that the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past (see Theorem 9 in this paper and [6, p.187 Lemma 9.21]). Suppose, that you want to use Lemma 9.21 with some events, that never occur, see as a comparison the paper [5] and the example for ST(1)={+∞} in the Summary. We don't have the element +1 in our above-mentioned time intervals [0, r[ and [0,+1[. This is only possible if we construct a new σ-algebra on R ∪ {−∞,+∞}. This construction is similar to the Borel-Sets and we call this σ-algebra extended Borel sets (see Definition 13 in this paper and [3, p. 21]). It can be proved, that {+∞} is an Element of extended Borel sets (see Theorem 21 in this paper). Now we use the interval [0,+∞] as a basis. We construct a σ-algebra on [0,+∞] similar to the book ([3, p. 12 18e]), see Definition 18 in this paper, and call it extended Borel subsets. We prove for stopping times with this given σ-algebra, that for τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we have the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past, see Theorem 25 in this paper. It is obvious, that {+∞} 2 extended Borel subsets. In general, Lemma 9.21 is important for the proof of the Optional Sampling Theorem, see 10.11 Proof of (i) in [6, p. 203]. [ABSTRACT FROM AUTHOR]

Optimal strategies are known for the finite and infinite horizon discrete-time search with constant unit cost and without recall. These strategies were obtained in the theory of optimal stopping, based on the martingale convergence theorem and other tools from probability theory. We present here an elementary approach to these problems, relying only on routine calculation of expected values. In the finite horizon case, the solution utilizes a simple form of backward induction, in conjunction with a nonlinear dynamical system, to compute the parameters of the optimal strategy. An elementary proof is also given that a simple threshold search is optimal among all strategies with finite expected total cost. [ABSTRACT FROM PUBLISHER]

We give under weak assumptions a complete combinatorial characterization of identifiability for linear mixtures of finite alphabet sources, with unknown mixing weights and unknown source signals, but known alphabet. This is based on a detailed treatment of the case of a single linear mixture. Notably, our identifiability analysis applies also to the case of unknown number of sources. We provide sufficient and necessary conditions for identifiability and give a simple sufficient criterion together with an explicit construction to determine the weights and the source signals for deterministic data by taking advantage of the hierarchical structure within the possible mixture values. We show that the probability of identifiability is related to the distribution of a hitting time and converges exponentially fast to one when the underlying sources come from a discrete Markov process. Finally, we explore our theoretical results in a simulation study. This paper extends and clarifies the scope of scenarios for which blind source separation becomes meaningful. [ABSTRACT FROM AUTHOR]

This article provides a two-stage procedure to develop a fixed-width confidence interval of log odds ratio in a joint binomial and inverse binomial setting where the stopping rule is obtained by adopting a two-stage procedure on the number of index subjects for the inverse sampling. A purely sequential version of this procedure is also studied. Different asymptotic results associated with the procedures are obtained. The findings are supported by detailed simulation study followed by one data example. [ABSTRACT FROM PUBLISHER]

. We start with the definition of stopping time according to [4], p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in [6], pp.37-38). Finally we give an example with some given filtration. Stopping time is very important for stochastic finance. A stopping time is the moment, where a certain event occurs ([7], p.372) and can be used together with stochastic processes ([4], p.283). Look at the following example: we install a function ST: {1,2,3,4} → {0, 1, 2} ∪ {+∞}, we define:. a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2. b. The set {0,1,2} consists of time points: 0=now, 1=tomorrow, 2=the day after tomorrow. We can prove: c. {w, where w is Element of Ω: ST.w=0}=ø & {w, where w is Element of Ω: ST.w=1}={1,2} & {w, where w is Element of Ω: ST.w=2}={3,4} and ST is a stopping time. We use a function Filt as Filtration of {0,1,2}, Σ where Filt(0)=Ωnow, Filt(1)=Ωfut1 and Filt(2)=Ωfut2. From a.,b. and c. we know that: d. {w, where w is Element of Ω: ST.w=0} in Ωnow and {w, where w is Element of Ω: ST.w=1} in Ωfut1 and {w, where w is Element of Ω: ST.w=2} in Ωfut2. The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also [7], p.371. Suppose we have ST(1)=+∞, then this means that for 1 the corresponding event never occurs. As an interpretation for our installed functions consider the given adapted stochastic process in the article [5]. ST(1)=1 means, that the given element 1 in {1,2,3,4} is stopped in 1 (=tomorrow). That tells us, that we have to look at the value f2(1) which is equal to 80. The same argumentation can be applied for the element 2 in {1,2,3,4}. ST(3)=2 means, that the given element 3 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(3) which is equal to 100. ST(4)=2 means, that the given element 4 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(4) which is equal to 120. In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [7], p.372). [ABSTRACT FROM AUTHOR]

We define and study order continuity, topological continuity, γ -Hölder-continuity and Kolmogorov–Čentsov-continuity of continuous-time stochastic processes in vector lattices and show that every such kind of continuous submartingale has a continuous compensator of the same kind. The notion of variation is introduced for continuous time stochastic processes and for a γ -Hölder-continuous martingale with finite variation, we prove that it is a constant martingale. The localization technique for not necessarily bounded martingales is introduced and used to prove our main result which states that the quadratic variation of a continuous-time γ -Hölder continuous martingale X is equal to its compensator 〈 X 〉 . [ABSTRACT FROM AUTHOR]

This paper considers the sequential composite hypothesis test with multiple sensors. The sensors observe random samples in parallel and communicate with a fusion center, who makes the global decision based on the sensor inputs. On the one hand, in the centralized scenario, where local samples are precisely transmitted to the fusion center, the generalized sequential likelihood ratio test (GSPRT) is shown to be asymptotically optimal in terms of the expected sample size as error rates tend to zero. On the other hand, for systems with limited power and bandwidth resources, decentralized solutions that only send a summary of local samples (we particularly focus on a one-bit communication protocol) to the fusion center is of great importance. To this end, we first consider a decentralized scheme where sensors send their one-bit quantized statistics every fixed period of time to the fusion center. We show that such a uniform sampling and quantization scheme is strictly suboptimal and its suboptimality can be quantified by the KL divergence of the distributions of the quantized statistics under both the hypotheses. We then propose a decentralized GSPRT based on level-triggered sampling. That is, each sensor runs its own GSPRT repeatedly and reports its local decision to the fusion center asynchronously. We show that this scheme is asymptotically optimal as the local thresholds and global thresholds grow large at different rates. Finally, two particular models and their associated applications are studied to compare the centralized and decentralized approaches. Numerical results are provided to demonstrate that the proposed level-triggered sampling based decentralized scheme aligns closely with the centralized scheme with substantially lower communication overhead, and significantly outperforms the uniform sampling and quantization-based decentralized scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider a size-dependent renewal risk model with stopping time claim-number process. In this model, we do not make any assumption on the dependence structure of claim sizes and inter-arrival times. We study large deviations of the aggregate amount of claims. For the subexponential heavy-tailed case, we obtain a precise large-deviation formula; our method substantially relies on a martingale for the structure of our models. [ABSTRACT FROM AUTHOR]

Although numerical methods of nonlinear stochastic differential delay equations (SDDEs) have been discussed by several authors, there is so far little work on the numerical approximation of SDDE with coefficients of polynomial growth. The main aim of the paper is to investigate convergence in probability of the Euler-Maruyama (EM) approximate solution for SDDE with one-sided polynomial growing drift coefficient and polynomial growing diffusion coefficient. Moreover, we prove the existence-and-uniqueness of almost surely exponentially stable global solution for this nonlinear stochastic delay system. Finally, a computer simulation confirms the efficiency of our numerical method. [ABSTRACT FROM AUTHOR]