Stochastic Ordering Research Articles

Stochastic ordering results on the duration of the gambler’s ruin game

AbstractIn the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a>0$ and $0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0<z\leq {a}/{2}$ , the family of distributions of the duration (total number of rounds) of the game indexed by $p \in [0,{\frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for ${a}/{2} \leq z<a$ , the family of distributions of the duration indexed by $p \in [{\frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for $z={a}/{2}$ , in terms of the likelihood ratio order, the distribution of the duration is maximized over $p \in [0,1]$ by $p={\frac{1}{2}}$ . The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.

Preservation of mean inactivity time ordering for coherent systems

AbstractPreservation of stochastic orders through the system signature has captured the attention of researchers in recent years. Signature-based comparisons have been made for the usual stochastic order, hazard rate order, and likelihood ratio orders. However, for the mean residual life (MRL) order, it has recently been proved that the preservation result does not hold true in general, but rather holds for a particular class of distributions. In this paper, we study whether or not a similar preservation result holds for the mean inactivity time (MIT) order. We prove that the MIT order is not preserved from signatures to system lifetimes with independent and identically distributed (i.i.d.) components, but holds for special classes of distributions. The relationship between these classes and the order statistics is also highlighted. Furthermore, the distribution-free comparison of the performance of coherent systems with dependent and identically distributed (d.i.d.) components is studied under the MIT ordering, using diagonal-dependent copulas and distorted distributions.

Two-parameter family of distributions: Properties, estimation, and applications

In this paper, a new flexible two-parameter family of distributions is introduced. Some mathematical properties, such as asymptotic behavior, moments, stochastic order, entropy, and quantile function, are examined. A special case of this family is introduced called Gemeay–Zeghdoudi distribution. The unknown parameters of the Gemeay–Zeghdoudi distribution are estimated via many estimators. The efficiency of the estimators is compared with a Monte Carlo simulation. The applicability of the new distribution is illustrated by the annual maximum flood and survival times of patients with breast cancer real data analyses.

A New Perspective on the Stochastic Fractional Order Materialized by the Exact Solutions of Allen-Cahn Equation

Stochastic fractional differential equations are among the most significant and recent equations in physical mathematics. Consequently, several scholars have recently been interested in these equations to develop analytical approximations. In this study, we highlight the stochastic fractional space Allen-Cahn equation (SFACE) as a major application of this class. In addition, we utilize the simplest equation method (SEM) with a dual sense of Brownian motion to convert the presented equation into an ordinary differential equation (ODE) and apply an effective computational technique to obtain exact solutions. By carefully comparing the derived solutions with solutions from other articles, we prove the distinction of these solutions for their diversity and the discovery of new solutions for SFACE that appear in many scientific fields, such as mathematical biology, quantum mechanics, and plasma physics. The results introduced in this article were obtained by plotting several graphs and examining how noise affects exact solutions using Mathematica and MATLAB software packages.

An Improved XShanker Distribution with Applications to Rainfall and Vinyl Chloride Data

This article is an improvement on the XShanker distribution having a single parameter which is the scale parameter and also in the class of Lindley distributions. It is named Double XShanker following the approach that generated it. The distributional properties which include the non-central moment with the associated statistics, the moment generating function, characteristic function, mean residual life function, stress-strength reliability function, Bonferroni and Lorenz curve functions, odd function, stochastic ordering, distribution of order statistics, and Reny entropy. The parameter was estimated using the method of maximum likelihood. Some visualizations based on theoretical values were presented. Some statistics were computed from theoretical values with convergence behavior recorded. A simulation study was conducted with 1000 samples of different sizes and the behavior observed is that as sample size increases the estimates decrease indicating precision. Data on rainfall and Vinyl chloride were used to validate the usefulness of the suggested distribution.

A New Member in the Lindley Class of Distributions with Flexible Applications

In this paper, a new lifetime distribution in the class of Lindley is developed and it is called "Doje Distribution". It is flexible in modeling lifetime data. The mathematical properties which include moments, the shape of the distribution, Quantile function, hazard function, survival function, stochastic ordering, mean deviation, Bonferroni and Lorenz curve, order statistic, and Renyi entropy have been studied. The maximum likelihood estimation has also been discussed. Two-lifetime data sets were utilized to demonstrate its usefulness. The results show that the proposed Doje distribution is better than Lindley, Ishita, Pareto, Chris-Jerry, Sushila, and Rama’s distribution in the instances of the data used.

Stress–Strength Reliability of a Failure Profile with Components Operating under the Internal Environmental Factors

In this paper, we study the stress–strength reliability of a failure profile in which, the components of the system are affected by the internal environmental factors. The internal environmental factors commonly change the random strength of a failure profile and have different effects on the random strength of profile components. Throughout this study, we investigated the effect of internal environmental factors under the various scenarios and computed the stress–strength reliability of a failure profile of order [Formula: see text]. Particularly, the stress–strength reliability of a [Formula: see text]-out-of-[Formula: see text] system has been evaluated based on several stochastic orders. In addition, the sensitivity analysis of the stress–strength reliability and the mean cost rate of failure profiles have been performed in the exponential model.

A quasi Rama distribution

A two-parameter quasi Rama distribution which contains the Rama distribution as particular case has been proposed. Its statistical properties based on moments have been discussed. The hazard rate function, mean residual life function, mean deviations and stochastic ordering of the distribution have been derived and studied. The estimation of parameters using method of moments and maximum likelihood methods has been discussed. A simulation study has been presented to know the performance of maximum likelihood estimates. The goodness of fit of the proposed distribution on two datasets relating to failure times has been presented.

Stochastic Comparisons of Largest-Order Statistics and Ranges from Marshall–Olkin Bivariate Exponential and Independent Exponential Variables

Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish some conditions for comparing the largest-order statistics (in the sense of mean residual life order) arising from bivariate Marshall–Olkin exponential distribution. Then, in the second part, we present some sufficient conditions for comparing sample ranges (in the sense of usual stochastic order and reversed hazard rate order) arising from independent exponential random variables.

Numerical simulation and analysis of the stochastic HIV/AIDS model in fractional order

The current work analyzes the HIV/AIDS model under stochastic fractional differential equations in the Caputo sense. The articulation of the model both in integer and arbitrary order with stochastic differential equation are presented. We provide the solution’s positivity and boundedness for the proposed system. The existence and uniqueness of the fractional stochastic order differential equations are given. We give a novel numerical approach based on Newton’s polynomial to numerically solve the fractional stochastic model. Also, we provide a numerical technique for the solution in the power law model. The results for the Caputo fractional case and stochastic fractional Caputo case are shown and discussed in detail. Sensitivity analysis are done and provide their respective results graphically. The numerical solution of the model based on their sensitive parameters is shown graphically in order to minimize the infection.

Stochastic comparisons of coherent systems with active redundancy at the component or system levels and component lifetimes following the accelerated life model

AbstractAn effective way to increase system reliability is to use redundancies (spares) into the systems either in component level or in system level. In this prospect, it is a significant issue that which set of available spares providing better system reliability in some stochastic sense. In this paper, we derive sufficient conditions under which a coherent system with a set of active redundancy at the component level or the system level provide better system reliability than that of the system with another set of redundancy, with respect some stochastic orders. We have derived the results for the component lifetimes following accelerated life (AL) model. The results obtained help us to design more reliable systems by allocating appropriate redundant components from the set of available options for the same. Various examples satisfying the sufficient conditions of the theoretical results are provided. Some results are illustrated with real‐world data.

Some Stochastic Orders over an Interval with Applications

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be adapted to the classical risk model in order to derive a moment bound for ruin probability. Finally, we compare it with other existing bounds.

Simplified models of remaining useful life based on stochastic orderings

Simplified models of remaining useful life based on stochastic orderings

Modified XLindley distribution: Properties, estimation, and applications

This article aims to introduce the inverse new XLindley distribution, a further extension of the new XLindley distribution. The article explores various properties of the proposed model, such as the quantile function, stochastic orders, entropies, fuzzy reliability, moments, and stress–strength estimation. The paper also compares different methods of estimating the parameters of the proposed model and evaluates their performance using a simulation study. Moreover, the paper demonstrates the usefulness of the proposed model by applying it to two real datasets. The article shows that the proposed model fits the data better than seven existing models based on model selection criteria, goodness-of-fit test statistics, and graphical visualizations. The paper concludes that the new model can be a valuable tool for modeling and analyzing hazard functions or survival data in various fields and contributing to probability theory and statistical inferences.

The Improved Stochastic Fractional Order Gradient Descent Algorithm

This paper mainly proposes some improved stochastic gradient descent (SGD) algorithms with a fractional order gradient for the online optimization problem. For three scenarios, including standard learning rate, adaptive gradient learning rate, and momentum learning rate, three new SGD algorithms are designed combining a fractional order gradient and it is shown that the corresponding regret functions are convergent at a sub-linear rate. Then we discuss the impact of the fractional order on the convergence and monotonicity and prove that the better performance can be obtained by adjusting the order of the fractional gradient. Finally, several practical examples are given to verify the superiority and validity of the proposed algorithm.

Stochastic Ordering Results on Implied Lifetime Distributions under a Specific Degradation Model

In this paper, a novel strategy is employed in which a degradation model affects the implied distribution of lifetimes differently compared to the traditional method. It is recognized that an existing link between the degradation measurements and failure time constructs the underlying time-to-failure model. We assume in this paper that the conditional survival function of a device under degradation is a piecewise linear function for a given level of degradation. The multiplicative degradation model is used as the underlying degradation model, which is often the case in many practical situations. It is found that the implied lifetime distribution is a classical mixture model. In this mixture model, the time to failure lies with some probabilities between two first passage times of the degradation process to reach two specified values. Stochastic comparisons in the model are investigated when the probabilities are changed. To illustrate the applicability of the results, several examples are given in cases when typical degradation models are candidates.

Characterizations of stochastic ordering for non-negative random variables

Characterizations of the usual stochastic order, hazard rate order, reversed hazard rate order and likelihood ratio order under weaker conditions than log-concavity were provided for non-negative random variables. Our results generalized the previous work in Yu (2009).

On [formula omitted]-out-of-[formula omitted] systems with homogeneous components and one independent cold standby redundancy

In this note, we study a k-out-of-n system with component lifetimes linked by Archimedean (survival) copula and independent of the lifetime of a cold standby redundancy. We derive the survival function and the residual life function of the redundant system. Also, we examine how the redundancy and the system structure impact the system reliability through developing the usual stochastic order on system lifetime. In particular, for parallel systems we investigate the role played by the component lifetime, the generator of Archimedean copula, and the starting time of the redundancy as well. The present framework well relates to the real situation and thus the main results are instructive to reliability engineers.

A modified power-law model: Properties, estimation, and applications

Abstract Models of various physical, biological, and artificial phenomena follow a power law over multiple magnitudes. This article presents a modified Pareto model with an upside-down shape and an adjustable right tail. The moments, quantiles, failure rate, mean residual life, and quantile residual life functions are examined. In addition, some stochastic ordering characteristics of the proposed model are investigated. The estimation of the parameters using the maximum likelihood estimator, the mean square error, and the Anderson–Darling estimator is explored, and a simulation study is conducted to analyze their behavior. Finally, we compare the proposed model with alternative methods for describing a dataset on the strength of carbon fiber and a dataset on customer waiting times in a bank.

The Type II Exponentiated Half Logistic-Marshall-Olkin-G Family of Distributions with Applications

A new generalized family of distributions called the type II exponentiated half logistic-Marshall-Olkin-G distribution is developed. Some special cases of the new model are presented. We explore some statistical properties of the new family of distributions. The statistical properties studied include expansion of the density function, hazard rate and quantile functions, moments, moment generating functions, probability weighted moments, stochastic ordering, distribution of order statistics and Rényi entropy. The maximum likelihood, ordinary and weighted least-squares techniques for the estimation of model parameters are presented, and Monte Carlo simulations for the new family of distributions are conducted. The importance of the new family of distributions is examined by means of applications to two real data sets.