Stochastic Monotonicity Research Articles

General weighted extropy of minimum and maximum ranked set sampling with unequal samples

. In industrial, environmental, and ecological investigations, ranked set sampling is a sample method that enables the experimenter to use the whole range of population values. The ranked set sampling process can be modified in two extremely helpful ways: maximum ranked set sampling with unequal samples and minimum ranked set sampling with unequal samples. They permit an increase in set size without too many ranking errors being introduced. In this article, we are defining general weighted extropy (GWJ) of minimum and maximum ranked set samples when samples are of unequal size (minRSSU and maxRSSU, respectively). Stochastic comparison and monotone properties have been studied under different situations. Additionally, we compare the extropy of these two sampling data with that of ranked set sampling data and simple random sampling data. Bounds of GWJ of minRSSU and maxRSSU have been obtained. Finally, we investigate the weighted discrimination information between simple random sampling, ranked set sampling, and minimum and maximum ranked set sampling with unequal sample sizes. Some results for equality of GWJ of minRSSU and maxRSSU under symmetric assumption are also obtained.

Modified Greenwood statistic and its application for statistical testing

In this paper, we explore the modified Greenwood statistic, which, in contrast to the classical Greenwood statistic, is properly defined for random samples from any distribution. The classical Greenwood statistic, extensively examined in the existing literature, has found diverse and interesting applications across various domains. Furthermore, numerous modifications to the classical statistic have been proposed. The modified Greenwood statistic, as proposed and discussed in this paper, shares several key properties with its classical counterpart. Emphasizing its stochastic monotonicity within three broad classes of distributions – namely, generalized Pareto, α−stable, and Student’s t distributions – we advocate for the utilization of the modified Greenwood statistic in testing scenarios. Our exploration encompasses three distinct directions. In the first direction, we employ the modified Greenwood statistic for Gaussian distribution testing. Our empirical results compellingly illustrate that the proposed approach consistently outperforms alternative goodness-of-fit tests documented in the literature, particularly exhibiting superior efficacy for small sample sizes. The second considered problem involves testing whether the distribution of a random sample has infinite variance. The last proposition suggests using the modified Greenwood statistic for testing of a given distribution. The presented simulation study strongly supports the efficiency of the proposed approach in the considered problems. Theoretical results and power simulation studies are further validated by real data analysis.

Parameter Estimation and Testing for the Doubly Geometric Process with Lognormal Distribution: Application to Bladder Cancer Patients’ Data

The geometric process (GP) has been widely utilized as a stochastic monotone model in the fields of probability and statistics. However, its practical application is often limited by certain assumptions. To address this, [Wu (2018). Doubly geometric process and applications. Journal of the Operational Research Society, 69(1), 66–67] introduced the doubly geometric process (DGP) as an extension of the GP model, relaxing some of its assumptions. Due to its ability to overcome the limitations of the GP model, the DGP has gained significant popularity in recent times. This study focuses on the parameter estimation problem for the DGP when the distribution of the first interarrival time follows a lognormal distribution with parameters [Formula: see text] and [Formula: see text]. We employ the maximum likelihood method to obtain estimates for both the model parameters and the distribution parameters. Additionally, we investigate the asymptotic joint distribution and statistical properties such as asymptotic unbiasedness and consistency of the estimators. Furthermore, we propose a novel test procedure to distinguish between the GP and DGP models. To assess the performance of the estimators and the proposed test procedure, we conduct a simulation study involving various sample sizes and parameter values. Finally, we present an application of the developed methods in fitting data from bladder cancer patients.

Stochastic monotone wing property: A new dependence structure for copulas

In this study, we introduce a novel concept of a copula dependence structure termed the stochastic monotone wing property and investigate its associated properties. This concept is closely related to the monotone conditional variance property, which describes the behavior of the conditional variance of V given U=u as either an increasing or a decreasing function of u for a given random vector (U,V) with a certain distribution function. The need for such a dependence structure arises in various fields, including insurance, where a copula family with the monotone conditional variance property is needed for the joint distribution of frequency and average severity in the collective risk model. Beyond insurance, its applications extend to economics and climatology. Furthermore, this study proposes a new parametric copula family that exhibits the stochastic monotone wing property and provides a method for constructing copulas with this characteristic.

Generalized BSDEs driven by RCLL martingales with stochastic monotone coefficients

A solution is given to generalized backward stochastic differential equations driven by a real-valued RCLL martingale on an arbitrary filtered probability space. The existence and uniqueness of a solution are proved via the Yosida approximation method when the generators are only stochastic monotone with respect to the y-variable and stochastic Lipschitz with respect to the z-variable, with different linear growth conditions.

On Stein’s method for stochastically monotone single-birth chains

We discuss Stein’s method for approximation by the stationary distribution of a single-birth Markov chain, in conjunction with stochastic monotonicity and similar assumptions. We use bounds on the increments of the solution of Poisson’s equation for such a process. Applications include rates of convergence to stationarity, and bounding the total variation distance between the stationary distributions of two Markov chains in the case where one transition matrix dominates the other.

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

Migration–contagion processes

AbstractConsider the following migration process based on a closed network of N queues with $K_N$ customers. Each station is a $\cdot$ /M/ $\infty$ queue with service (or migration) rate $\mu$ . Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate $\alpha Y$ if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate $\beta$ . We let N tend to infinity and assume that $\lim_{N\to \infty} K_N/N= \eta $ , where $\eta$ is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to $\eta$ . The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.

Tie-Break Bootstrap for Nonparametric Rank Statistics

In this article, we propose a new bootstrap procedure for the empirical copula process. The procedure involves taking pseudo samples of normalized ranks in the same fashion as the classical bootstrap and applying small perturbations to break ties in the normalized ranks. Our procedure is a simple modification of the usual bootstrap based on sampling with replacement, yet it provides noticeable improvement in the finite sample performance. We also discuss how to incorporate our procedure into the time series framework. Since nonparametric rank statistics can be treated as functionals of the empirical copula, our proposal is useful in approximating the distribution of rank statistics in general. As an empirical illustration, we apply our bootstrap procedure to test the null hypotheses of positive quadrant dependence, tail monotonicity, and stochastic monotonicity, using U.S. Census data on spousal incomes in the past 15 years.

Optimal algorithms for differentially private stochastic monotone variational inequalities and saddle-point problems

In this work, we conduct the first systematic study of stochastic variational inequality (SVI) and stochastic saddle point (SSP) problems under the constraint of differential privacy (DP). We propose two algorithms: Noisy Stochastic Extragradient (NSEG) and Noisy Inexact Stochastic Proximal Point (NISPP). We show that a stochastic approximation variant of these algorithms attains risk bounds vanishing as a function of the dataset size, with respect to the strong gap function; and a sampling with replacement variant achieves optimal risk bounds with respect to a weak gap function. We also show lower bounds of the same order on weak gap function. Hence, our algorithms are optimal. Key to our analysis is the investigation of algorithmic stability bounds, both of which are new even in the nonprivate case. The dependence of the running time of the sampling with replacement algorithms, with respect to the dataset size n, is n2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^2$$\\end{document} for NSEG and O~(n3/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{O}}(n^{3/2})$$\\end{document} for NISPP.

Transform orders and stochastic monotonicity of statistical functionals

AbstractIn some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such a behavior in terms of the usual stochastic order. For this purpose, we introduce a generalized family of stochastic orders, which is referred to as transform orders, showing that it provides a flexible framework for deriving stochastic monotonicity results. Given that our general definition makes it possible to obtain some well known ordering relations as particular cases, we can easily apply our method to different families of functionals. These include some prominent inequality measures, such as the generalized entropy, the Gini index, and its generalizations. We also illustrate the applicability of our approach by determining the least favorable distribution, and the behavior of some bootstrap statistics, in some goodness‐of‐fit testing procedures.

Travelling wave of stochastic Lotka-Volterra competitive system

<p style='text-indent:20px;'>This work is devoted to the travelling wave solutions and asymptotic wave speed for a two-species stochastic competitive system driven by a multiplicative Itô type white noise. By the introduction of sup-solutions and sub-solutions, the existence and the approximations of the travelling wave solutions are proven under some conditions on the parameters. Moreover, combining with the stochastic Feynman-Kac formula, sub-solution, sup-solution, monotone iteration and equilibria of the kinetic system corresponding to the transformed stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.</p>

Empirical Reconstruction of Earth‐Moon and Solar System Dynamical Parameters for the Past 2.5 Billion Years From Cyclostratigraphy

AbstractThe chaotic behavior of the Solar System prevents the prediction of the precise evolution of the planetary secular frequencies. Here we use four cyclostratigraphic sequences, ∼259 Ma Wujiaping Formation, ∼455 Ma Pingliang Formation, ∼655 Ma Datangpo Formation and ∼2,465 Ma Dales Gorge Member, and Bayesian inversion to provide powerful geological constraints to this problem. The inversion provides constraining estimates for the Solar System fundamental secular frequencies g1, g2, g3, g4, and g5, and the Earth's precession rate k from 105.26 ± 2.68 (2σ) arcseconds/year at ∼2,465 Ma to 55.86 ± 2.60 (2σ) arcseconds/year at ∼259 Ma. Stochastic monotone interpolation enables the reconstruction of Earth‐Moon distance from 2.5 billion years to the present. Calculation of tidal dissipation supports the ocean model before 1.4 Ga, but is outside the error range after ∼1.8 Ga, suggesting lower tidal dissipation in the early Proterozoic Eon, and a possible high value in the early Paleozoic.

On Generalized Reflected BSDEs with Rcll Obstacle

Abstract This paper proves the existence and uniqueness theorem for generalized (reflected) backward stochastic differential equations under stochastic Lipschitz and monotone condition. The result is shown by using Picard’s iteration, the Snell envelope theory and the penalization method.

Nonparametric inference of complier quantile treatment effects in randomized trials with imperfect compliance

To analyze randomized trials with imperfect compliance, a standard approach is to estimate the local average treatment effect in the sub-population of compliers using randomization status as an instrumental variable. Though quantile analysis has been popular in general, the local (or complier) quantile treatment effect (cQTE) as a causal estimand has received insufficient attention. In this paper, we map out the details for the estimation, inference, and sensitivity analysis of the cQTE in a completely nonparametric setting. We propose to estimate the cQTE using nonparametric plug-in estimators of the cumulative distribution functions for the potential outcomes of the compliers. The cQTE estimator is shown to be asymptotically normal, with asymptotic variance estimated through kernel-smoothed density estimators. The procedure is easily extended to adjust for discrete covariates for gains in statistical efficiency. Moreover, by exploiting the stochastic monotonicity of the quantile functional, we develop sensitivity bounds for the cQTE when key assumptions such as exclusion restriction and instrument monotonicity are violated. Extensive simulations show that the proposed methods provide valid inference of the target local estimand and outperform standard intent-to-treat tests, especially under low compliance rates and/or heterogeneous treatment effects. A recent study on a government-funded health insurance program in India is analyzed as an illustration.

New Stochastic Orders and Monotone Comparative Statics of Changes in Risk under Risk Aversion

New Stochastic Orders and Monotone Comparative Statics of Changes in Risk under Risk Aversion

Statistical inference for doubly geometric process with exponential distribution

The geometric process is widely applied as a stochastic monotone model in many practical applications since its introduction. However, it sometimes does not satisfy some requirements in the real-world applications due to model limitations. For this reason, it is proposed a new stochastic model which is called doubly geometric process. In the applications of the doubly geometric process, the estimation problem associated with the process arises naturally. In this study, the statistical inference problem for the doubly geometric process is considered by assuming that the distribution of the first interarrival time has an exponential distribution. The maximum likelihood method is used to estimate the model parameters of the doubly geometric process and the parameter of distribution. The joint distribution of the maximum likelihood estimators is obtained. A simulation study is presented to evaluate the small sample performance of the estimators with different parameter values. Finally, three real-world-data sets are used to illustrate the applicability of the methods.

Stochastic monotonicity of dependent variables given their sum

Given a finite set of independent random variables, assume one can observe their sum, and denote with s its value. Efron in 1965, and Lehmann in 1966, described conditions on the involved variables such that each of them stochastically increases in the value s, i.e., such that the expected value of any non-decreasing function of the variable increases as s increases. In this paper, we investigate conditions such that this stochastic monotonicity property is satisfied when the assumption of independence is removed. Comparisons in the stronger likelihood ratio order are considered as well.

On the reliability estimation of stochastic binary systems

AbstractA stochastic binary system (SBS) is a multicomponent on‐off system subject to random independent failures on its components. After potential failures, the state of the subsystem is ruled by a logical function (called structure function) that determines whether the system is operational or not. A SBS serves as a natural generalization of network reliability analysis, where the goal is to find the probability of correct operation of the system (in terms of connectivity, network diameter, or different measures of success). A particular subclass of interest is stochastic monotone binary systems (SMBS), which are characterized by nondecreasing structure. We explore the combinatorics of SBS, which provide building blocks for system reliability estimation, looking at minimal nonoperational subsystems, called mincuts. One key concept to understand the underlying combinatorics of SBS is duality. As methods for exact evaluation take exponential time, we discuss the use of Monte Carlo algorithms. In particular, we discuss the F‐Monte Carlo method for estimating the reliability polynomial for homogeneous SBS, the recursive variance reduction for SMBS, which builds upon the efficient determination of mincuts, and three additional methods that combine in different ways the well‐known techniques of permutation Monte Carlo and splitting. These last three methods are based on a stochastic process called the creation process, a temporal evolution of the SBS which is static by definition. All the methods are compared using different topologies, showing large efficiency gains over the basic Monte Carlo scheme.

Stochastic monotonicity and the Markov product for copulas

Given two random variables X and Y, stochastic monotonicity describes a monotone influence of X on Y. We prove two different characterizations of stochastically monotone 2-copulas using the isomorphism between 2-copulas and Markov operators. The first approach establishes a one-to-one correspondence between stochastically monotone copulas and monotonicity-preserving Markov operators. The second approach characterizes stochastically monotone copulas by their monotonicity property with respect to the Markov product. Applying the latter result, we identify all idempotent stochastically monotone copulas as ordinal sums of the independence copula Π.