This paper probabilistically explores a class of stationary count time series models built by superpositioning (or otherwise combining) independent copies of a binary stationary sequence of zeroes and ones. Superpositioning methods have proven useful in devising stationary count time series having prespecified marginal distributions. Here, basic properties of this model class are established and the idea is further developed. Specifically, stationary series with binomial, Poisson, negative binomial, discrete uniform, and multinomial marginal distributions are constructed; other marginal distributions are possible. Our primary goal is to derive the autocovariance function of the resulting series.

The principle of maximum entropy is a well-known approach to produce a model for data-generating distributions. In this approach, if partial knowledge about the distribution is available in terms of a set of information constraints, then the model that maximizes entropy under these constraints is used for the inference. In this paper, we propose a new three-parameter lifetime distribution using the maximum entropy principle under the constraints on the mean and a general index. We then present some statistical properties of the new distribution, including hazard rate function, quantile function, moments, characterization, and stochastic ordering. We use the maximum likelihood estimation technique to estimate the model parameters. A Monte Carlo study is carried out to evaluate the performance of the estimation method. In order to illustrate the usefulness of the proposed model, we fit the model to three real data sets and compare its relative performance with respect to the beta generalized Weibull family.

The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

Let X be a random variable with nonsymmetric density p(x). We give the symmetric density q(x) closest to it in the sense of Kulback–Liebler and Hellinger distances. (All symmetries are around zero.) For the first distance, we show that q(x) is proportional to the geometric mean of p(x) and p(−x). For example, a symmetrized shifted exponential is a centered uniform, and a symmetrized shifted gamma is a centered beta random variable. For the second distance, q(x) is proportional to the square of the arithmetic mean of p(x)1/2 and p(−x)1/2. Sample versions are also given for each. We also give the optimal random function f such that f(X) is symmetrically distributed and minimizes |f(X)−X|. Finally, we show how to optimize the Hellinger distance for vector X subject to supersymmetry and for scalar X subject to being monotone about zero in each half-line.

Recently, the asymptotic mean value of the height for a birth-and-death process is given in Videla [Videla, L.A. (2020)]. We consider the asymptotic variance of the height in the case when the number of states tends to infinity. Further, we prove that the heights exhibit a cutoff phenomenon and that the normalized height converges to a degenerate distribution.

This paper addresses a joint pricing and inventory control problem for a batch production system with random leadtimes. Assume that demand arrives according to a Poisson process with a price-dependent arrival rate. Each replenishment order contains a single batch of a fixed lot size. The replenishment leadtime follows an Erlang distribution, with the number of completed phases recording the delivery state of outstanding orders. The objective is to determine an optimal inventory-pricing policy that maximizes total expected discounted profit or long-run average profit. We first show that when there is at most one order outstanding at any point in time and that excess demand is lost, the optimal reorder policy can be characterized by a critical stock level and the optimal pricing decision is decreasing in the inventory level and delivery state. We then extend the analysis to mixed-Erlang leadtime distribution which can be used to approximate any random leadtime to any degree of accuracy. We further extend the analysis to allowing three outstanding orders where the optimal reorder point becomes state-dependent: the closer an outstanding order is to its arrival or the more orders are outstanding, the lower selling price is charged and the lower reorder point is chosen. Finally, we address the backlog case and show that the monotone pricing structure may not be true when the optimal reorder point is negative.

This article investigates TP2 dependence of sample spacings. It is proved that TP2 (RR2) dependence between a general spacing and a nonadjacent order statistic might be characterized by the DLR (ILR) property of the parent distribution, and TP2 dependence between any pair of consecutive spacings might be characterized by the DLR aging property of the population. Furthermore, TP2 dependence between any two consecutive spacings in multiple outliers exponential models is also derived. In addition, some applications in reliability and business auction are presented as well.

A cascading failure is a failure in a system of interconnected parts, in which the breakdown of one element can lead to the subsequent collapse of the others. The aim of this paper is to introduce a simple combinatorial model for the study of cascading failures. In particular, having in mind particle systems and Markov random fields, we take into consideration a network of interacting urns displaced over a lattice. Every urn is Pólya-like and its reinforcement matrix is not only a function of time (time contagion) but also of the behavior of the neighboring urns (spatial contagion), and of a random component, which can represent either simple fate or the impact of exogenous factors. In this way a non-trivial dependence structure among the urns is built, and it is used to study default avalanches over the lattice. Thanks to its flexibility and its interesting probabilistic properties, the given construction may be used to model different phenomena characterized by cascading failures such as power grids and financial networks.

Let X1, …, Xn be non-negative, independent and identically distributed random variables with a common distribution function F, and denote by X1:n ≤ ··· ≤ Xn:n the corresponding order statistics. In this paper, we investigate the second-order regular variation (2RV) of the tail probabilities of Xk:n and Xj:n − Xi:n under the assumption that $\bar {F}$ is of the 2RV, where 1 ≤ k ≤ n and 1 ≤ i < j ≤ n.

Consider an absolutely continuous distribution on [0, ∞) with known mean μ, and hazard rate function, h satisfying, 0<a≤h(t)≤b<∞, for almost all t≥0. We derive the sharp range for σ2, under these constraints.

Much of the work relating to control charts is on monitoring observations for changes in mean. Here, for the first time, we develop a class of cusum-type statistics to monitor any aspect of a process (for example, mean, variance, skewness or kurtosis). This development is based on recently published papers by the authors.

This article offers an approach for studying the time-dependent occupancy distribution for a modest generalization of the GI/G/1 queuing system in which interarrival times and service times, although mutually independent, are not necessarily identically distributed. We develop and explore an analytical model leading to a computational approach that gives tight bounds on the occupancy distribution. Although there is no general closed-form characterization of probability law dynamics for occupancy in the GI/G/1 queue, our results offer what might be termed “near-closed-form” in that accurate plots of the transient occupancy distribution can be constructed with an insignificant computational burden. We believe that our results are unique; we are unaware of any alternative analytical approach leading to a numerical characterization of the time-dependent occupancy distribution for the G/G/1 queuing systems considered here.

Our analyses employ a marked point process that converges to the occupancy process at any fixed time t; it is shown that this process forms a Markov chain from which the transient occupancy law is available. We verify our analytical approach via comparison with the well-known closed-form expressions for time-dependent occupancy distribution of the M/M/1 queue. Additionally, we suggest the viability of our approach, as a computational means of obtaining the time-dependent occupancy distribution, through straightforward application to a Gamma[x]/Weibull/1 queuing system having batch arrivals and batch job services.

In this paper, we introduce a novel way to quantify the remaining inaccuracy of order statistics by utilizing the concept of extropy. We explore various properties and characteristics of this new measure. Additionally, we expand the notion of inaccuracy for ordered random variables to a dynamic version and demonstrate that this dynamic information measure provides a unique determination of the distribution function. Moreover, we investigate specific lifetime distributions by analyzing the residual inaccuracy of the first-order statistics. Nonparametric kernel estimation of the proposed measure is suggested. Simulation results show that the kernel estimator with bandwidth selection using the cross-validation method has the best performance. Finally, an application of the proposed measure on the model selection is provided.

We consider a variation of the processor-sharing (PS) queue, inspired by freelance job websites where multiple freelancers compete for a single job. We develop fluid limit approximations for the overloaded PS-model with multiple (possibly infinitely many) service stages. Based on this approximation, we estimate what proportion of freelancers get the job they apply for. In addition, the PS model studied here is an instance of PS with routing and impatience, for which no Lyapunov function is known, and we suggest some partial solutions.

A second-order continuous-state-space model for two-dimensional queueing systems is developed in this article. A particular problem is treated but the results can apply to some other two-dimensional queueing problems directly. The generating function for the model is obtained by applying a Riemann boundary value problem and leads to a computationally feasible solution.

As the formal methods of proving correctness of a computer program are still very inadequate, in practice when a new piece of software is developed and all obvious errors are removed, it is tested with different (random) inputs in order to detect the remaining errors and assess its quality. We suppose that whenever the program fails the error causing the failure can be detected and removed correctly. Thus, the quality of the software increases as testing goes on. In this paper, we consider two different models and present the minimum variance unbiased estimators of the expected failure rate of the revised software at any time of testing t, based on the data generated up to that point.

We consider a system with heterogeneous unreliable components that requires only one component to be turned on in order for it to operate. Repair workers may have different skills and may be unavailable for random periods of time. The problem is to determine a usage and repair policy to maximize system availability. We give conditions under which the optimal usage policy is to always use, or turn on, the component with the shortest repair time, and the optimal repair policy is to always repair the most reliable component (with the smallest failure rate). We fully characterize the optimal policy when there are only two components. Our system is equivalent to a closed system with multiple single-server queues, where the objective is to minimize server idle time at one of the queues.

We discuss a new stochastic ordering for the sequence of independent random variables. It generalizes the stochastic precedence (SP) order that is defined for two random variables to the case n > 2. All conventional stochastic orders are transitive, whereas the SP order is not. Therefore, a new approach to compare the sequence of random variables had to be developed that resulted in the notion of the sequential precedence order. A sufficient condition for this order is derived and some examples are considered.

We consider a general polling model with N stations. The stations are served exhaustively and in cyclic order. Once a station queue falls empty, the server does not immediately switch to the next station. Rather, it waits at the station for the possible arrival of new work (“wait-and-see”) and, in the case of this happening, it restarts service in an exhaustive fashion. The total time the server waits idly is set to be a fixed, deterministic parameter for each station. Switchover times and service times are allowed to follow some general distribution, respectively. In some cases, which can be characterized, this strategy yields a strictly lower average queuing delay than for the exhaustive strategy, which corresponds to setting the “wait-and-see credit” equal to zero for all stations. This extends the results of Peköz [12] and of Boxma et al. [4]. Furthermore, we give a lower bound for the delay for all strategies that allow the server to wait at the stations even though no work is present.