Distribution Of Stopping Time Research Articles

Using Large-Scale Drone Data to Monitor and Assess the Behavior of Freight Vehicles on Urban Level

Compared with public transport operations, urban freight traffic and its associated delivery operations seem to be frequently overlooked in urban traffic management and traffic flow theory. One explanation for this is certainly the lack of available data, as the competitive freight transport market is fragmented and several actors are unwilling to collect or share tactical and operational data. In this study, we use the unique pNEUMA drone data set from Athens, Greece, to shed light on urban freight operations. We discuss macroscopic traffic indicators in a multimodal context. As the vehicle stopping behavior can adversely influence traffic flow, we reveal the stopping behavior of the different modes represented in the data set using clustering techniques. We find that urban freight vehicles’ stopping frequency lies between the stopping frequencies of cars and buses. We reveal the distribution of stopping times for loading and unloading stops in Athens to have a mean of around 380 seconds. Clustering all loading and unloading stops further reveals three groups of loading and unloading stops that could be labeled by incorporating knowledge and expertise about local particularities. The limited flight time of drones, owing to their battery capacities, did not allow reconstruction of longer vehicle routes, such as an entire vehicle tour within the network. However, this could be addressed in future research by realizing continuous large-scale monitoring routines. The revealed vehicle behavior parameters can be used in traffic models to generate further insights into the impacts of urban freight transport to inform public sector decision makers.

Graph planning with expected finite horizon

Fixed-horizon planning considers a weighted graph and asks to construct a path that maximizes the sum of weights for a given time horizon T. However, in many scenarios, the time horizon is not fixed, but the stopping time is chosen according to some distribution such that the expected stopping time is T. If the stopping-time distribution is not known, then to ensure robustness, the distribution is chosen by an adversary as the worst-case scenario. A stationary plan for every vertex always chooses the same outgoing edge. For fixed horizon or fixed stopping-time distribution, stationary plans are not sufficient for optimality. Quite surprisingly we show that when an adversary chooses the stopping-time distribution with expected stopping-time T, then stationary plans are sufficient. While computing optimal stationary plans for fixed horizon is NP-complete, we show that computing optimal stationary plans under adversarial stopping-time distribution can be achieved in polynomial time.

Non-Markovian random walks characterize network robustness to nonlocal cascades

Localized perturbations in a real-world network have the potential to trigger cascade failures at the whole system level, hindering its operations and functions. Standard approaches analytically tackling this problem are mostly based either on static descriptions, such as percolation, or on models where the failure evolves through first-neighbor connections, crucially failing to capture the nonlocal behavior typical of real cascades. We introduce a dynamical model that maps the failure propagation across the network to a self-avoiding random walk that, at each step, has a probability to perform nonlocal jumps toward operational systems' units. Despite the inherent non-Markovian nature of the process, we are able to characterize the critical behavior of the system out of equilibrium, as well as the stopping time distribution of the cascades. Our numerical experiments on synthetic and empirical biological and transportation networks are in excellent agreement with theoretical expectation, demonstrating the ability of our framework to quantify the vulnerability to nonlocal cascade failures of complex systems with interconnected structure.

Prophet secretary through blind strategies

In the classic prophet inequality, a well-known problem in optimal stopping theory, samples from independent random variables (possibly differently distributed) arrive online. A gambler who knows the distributions, but cannot see the future, must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet that sees all the realizations in advance gets. In the late seventies, Krengel and Sucheston (Bull Am Math Soc 83(4):745–747, 1977), established that this worst case ratio is 0.5. A particularly interesting variant is the so-called prophet secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms are known to achieve a constant of $$1-1/e \approx 0.632$$ and very recently this barrier was slightly improved by Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). In this paper we introduce a new type of multi-threshold strategy, called blind strategy. Such a strategy sets a nonincreasing sequence of thresholds that depends only on the distribution of the maximum of the random variables, and the gambler stops the first time a sample surpasses the threshold of the stage. Our main result shows that these strategies can achieve a constant of 0.669 for the prophet secretary problem, improving upon the best known result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018), and even that of Beyhaghi et al. (Improved approximations for posted price and second price mechanisms. CoRR arXiv:1807.03435 , 2018) that works in the case in which the gambler can select the order of the samples. The crux of the result is a very precise analysis of the underlying stopping time distribution for the gambler’s strategy that is inspired by the theory of Schur-convex functions. We further prove that our family of blind strategies cannot lead to a constant better than 0.675. Finally we prove that no algorithm for the gambler can achieve a constant better than $$\sqrt{3}-1 \approx 0.732$$ , which also improves upon a recent result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). This implies that the upper bound on what the gambler can get in the prophet secretary problem is strictly lower than what she can get in the i.i.d. case. This constitutes the first separation between the prophet secretary problem and the i.i.d. prophet inequality.

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.

Timing portfolio strategies with exponential Lévy processes

This paper analyses the impact of parametric timing portfolio strategies on the U.S. stock market. In particular, we assume that the log-returns follow a given parametric Levy process and we describe a methodology to approximate the distributions of stopping times using the underlying Markov transition matrix. Therefore, we propose the use of portfolio strategies based on the maximization of the ratio between the expected first passage time to reach a low level of wealth and the expected first passage time to reach a high level of wealth. Finally, we compare the ex-post wealth obtained maximizing the ratio of proper expected stopping times under different distributional assumptions.

Distribution‐constrained optimal stopping

AbstractWe solve the problem of optimal stopping of a Brownian motion subject to the constraint that the stopping time's distribution is a given measure consisting of finitely many atoms. In particular, we show that this problem can be converted to a finite sequence of state‐constrained optimal control problems with additional states corresponding to the conditional probability of stopping at each possible terminal time. The proof of this correspondence relies on a new variation of the dynamic programming principle for state‐constrained problems, which avoids measurable selections. We emphasize that distribution constraints lead to novel and interesting mathematical problems on their own, but also demonstrate an application in mathematical finance to model‐free superhedging with an outlook on volatility.

Finite-Length Analysis of BATS Codes

BATS codes were proposed for communication through networks with packet loss. A BATS code consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code, which works with the inner code that comprises random linear coding at the intermediate network nodes. In this paper, the performance of finite-length BATS codes is analyzed with respect to both belief propagation (BP) decoding and inactivation decoding. Our results enable us to evaluate efficiently the finite-length performance in terms of the number of batches used for decoding ranging from 1 to a given maximum number, and provide new insights on the decoding performance. Specifically, for a fixed number of input symbols and a range of the number of batches used for decoding, we obtain recursive formulae to calculate the stopping time distribution of BP decoding and the inactivation probability in inactivation decoding. We also find that both the failure probability of BP decoding and the expected number of inactivations in inactivation decoding can be expressed in a power-sum form where the number of batches appears only as the exponent. This power-sum expression reveals clearly how the decoding failure probability and the expected number of inactivation decrease with the number of batches. When the number of batches used for decoding follows a Poisson distribution, we further derive recursive formulas with potentially lower computational complexity for both decoding algorithms. For the BP decoder that consumes batches one by one, three formulae are provided to characterize the expected number of consumed batches until all the input symbols are decoded.

Analogues of the 3 x + 1 Problem in Polynomial Rings of Characteristic 2

ABSTRACTThe Collatz conjecture (also known as the 3x + 1 problem) concerns the behavior of the discrete dynamical system on the positive integers defined by iteration of the so-called 3x + 1 function. We investigate analogous dynamical systems in rings of functions of algebraic curves over . We prove in this setting a generalized analogue of a theorem of Terras concerning the asymptotic distribution of stopping times. We also present experimental data on the behavior of these dynamical systems.

On Rates of Convergence for Markov Chains Under Random Time State Dependent Drift Criteria

Many applications in networked control require intermittent access of a controller to a system, as in event-triggered systems or information constrained control applications. Motivated by such applications and extending previous work on Lyapunov-theoretic drift criteria, we establish both subgeometric and geometric rates of convergence for Markov chains under state dependent random time drift criteria. We quantify how the rate of ergodicity, nature of Lyapunov functions, their drift properties, and the distributions of stopping times are related. We finally study an application in networked control.

AN ANALYTICAL SOLUTION FOR THE TWO‐SIDED PARISIAN STOPPING TIME, ITS ASYMPTOTICS, AND THE PRICING OF PARISIAN OPTIONS

In this paper, we obtain a recursive formula for the density of the two‐sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one‐sided Parisian stopping time derived in Dassios and Lim. However, when we study the tails of the two distributions, we find that the two‐sided stopping time has an exponential tail, while the one‐sided stopping time has a heavier tail. We derive an asymptotic result for the tail of the two‐sided stopping time distribution and propose an alternative method of approximating the price of the two‐sided Parisian option.

Reaction times of monitoring schemes for ARMA time series

This paper is concerned with deriving the limit distributions of stopping times devised to sequentially uncover structural breaks in the parameters of an autoregressive moving average, ARMA, time series. The stopping rules are defined as the first time lag for which detectors, based on CUSUMs and Page's CUSUMs for residuals, exceed the value of a prescribed threshold function. It is shown that the limit distributions crucially depend on a drift term induced by the underlying ARMA parameters. The precise form of the asymptotic is determined by an interplay between the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications to electroencephalography, EEG, and IBM data. The empirical results indicate a satisfactory behavior in finite samples.

Distributions of stopping times in some sequential estimation procedures

A class of sequential estimation procedures is considered in the case when relevant data may become available only at random times. The exact distributions of the optimal stopping time and the number of observations at the moment of stopping are derived in some sequential procedures. The results obtained in an explicit form are applied to derive the expected time of observing the process, the average number of observations and the expected loss of sequential estimation procedures based on delayed observations. The use of the results is illustrated in a special model of normally distributed observations and the Weibull distributed lifetimes. The probabilistic characteristics are also derived for an adaptive sequential procedures and the behavior of the adaptive procedure is compared with the corresponding optimal sequential procedure.

A comparison of two worlds: How does Bayes hold up to the status quo for the analysis of clinical trials?

Background There is a paucity of literature comparing Bayesian analytic techniques with traditional approaches for analyzing clinical trials using real trial data. Methods We compared Bayesian and frequentist group sequential methods using data from two published clinical trials. We chose two widely accepted frequentist rules, O'Brien–Fleming and Lan–DeMets, and conjugate Bayesian priors. Using the nonparametric bootstrap, we estimated a sampling distribution of stopping times for each method. Because current practice dictates the preservation of an experiment-wise false positive rate (Type I error), we approximated these error rates for our Bayesian and frequentist analyses with the posterior probability of detecting an effect in a simulated null sample. Thus for the data-generated distribution represented by these trials, we were able to compare the relative performance of these techniques. Results No final outcomes differed from those of the original trials. However, the timing of trial termination differed substantially by method and varied by trial. For one trial, group sequential designs of either type dictated early stopping of the study. In the other, stopping times were dependent upon the choice of spending function and prior distribution. Conclusions Results indicate that trialists ought to consider Bayesian methods in addition to traditional approaches for analysis of clinical trials. Though findings from this small sample did not demonstrate either method to consistently outperform the other, they did suggest the need to replicate these comparisons using data from varied clinical trials in order to determine the conditions under which the different methods would be most efficient.

Sequential Testing to Guarantee the Necessary Sample Size in Clinical Trials

This article addresses the sequential sampling issues related to attainment of a given number of subjects in a Poisson arrival process. The problem of guaranteeing the necessary sample size is formulated as that of open-ended hypothesis testing. To implement the procedure a one-sided stopping boundary must be specified. The formulas for the probabilities of general stopping time distribution are derived. For the linear boundary they are in the class of Lagrangian Poisson distributions. A locally optimal test is obtained.

Analytic Combinatorics of the Mabinogion Urn

The Mabinogion urn is a simple model of the spread of influences amongst versatile populations. It corresponds to a non-standard urn with balls of two colours: each time a ball is drawn, it causes a ball of the other kind to switch its colour. The process stops once unanimity has been reached. This note provides analytic expressions describing the evolution of the Mabinogion urn, based on a time-reversal transformation applied to the classical Ehrenfest urn. Consequences include a precise asymptotic analysis of the stopping-time distribution―it is asymptotically normal in the "unfair'' case and akin to an extreme-value (double exponential) distribution in the "fair'' case―as well as a characterization of the exponentially small probability of reversing a majority.

Authors’ response

In this rejoinder, we address all or almost all comments that have been provided by the discussants. In particular, we include additional references to literature related to change-point detection as well as to intrusion detection; analyze distributions of stopping times of the CUSUM and Shiryaev–Roberts detection procedures under the no change hypothesis in more detail; and provide an overview of the detection-isolation problem.

Some recent results on the distributions of stopping times of compound Poisson processes with linear boundaries

A review of recent results is given, for the distribution of stopping times defined on compound Poisson processes and linear boundaries. Generalization of the results of Zacks (Commun. Statist. Stochastic Models 7 (1991) 233) is given for discrete compound Poisson processes. The main approach in the reviewed papers is that of sample path analysis, which leads to renewal type equations. Explicit solutions are given.

Multihypothesis sequential probability ratio tests. II. Accurate asymptotic expansions for the expected sample size

For pt. I see ibid. vol.45, p.2448-61, 1999. We proved in pt.I that two specific constructions of multihypothesis sequential tests, which we refer to as multihypothesis sequential probability ratio tests (MSPRTs), are asymptotically optimal as the decision risks (or error probabilities) go to zero. The MSPRTs asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution, under very general statistical models for the observations. In this paper, based on nonlinear renewal theory we find accurate asymptotic approximations (up to a vanishing term) for the expected sample size that take into account the "overshoot" over the boundaries of decision statistics. The approximations are derived for the scenario where the hypotheses are simple, the observations are independent and identically distributed (i.i.d.) according to one of the underlying distributions, and the decision risks go to zero. Simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes. The asymptotic results given here complete the analysis initiated by Baum and Veeravalli (1994), where first-order asymptotics were obtained for the expected sample size under a specific restriction on the Kullback-Leibler distances between the hypotheses.

Distributions of stopping times for compound poisson processes with positive jumps and linear boundaries

Let be a compound Poisson process, i.e. positive random variables and a time homogeneous Poisson jump process. We consider two linear boundaries and and the stopping times and . Laplace Stieltjes transforms are developed for the distributions of TU and T for general distribution of . These transforms are obtained by analyzing the sample path behavior of the process {Yt }. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more