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This study considers a tandem supply chain (S.C.) in which a credit-worthy retailer cooperates with a capital-constrained supplier to manufacture products. The retailer can use two major pre-shipment financing options, either cash in advance (CIA) or pricing support purchase order financing (PSPOF), to fund the supplier's operations. A model is developed to examine the interactions among the entities. The retailer can choose the prepayment modality, purchase price and order quantity. The supplier decides on the production input and loan amount. A Karush-Kuhn-Tucker optimality analysis is used to characterise the properties of the retailer's ordering and financing strategy: (1) The supplier might embezzle the CIA if the purchasing and financing contract is not well-designed; therefore, the retailer should establish some restrictions on the order price and CIA amount. (2) The retailer's optimal decision about the quantity to purchase is independent of the financing strategy. (3) Because these schemes are interchangeable, multiple optimal financing portfolios consisting of CIA and/or PSPOF are possible. (4) The optimal dual-financing scheme maximises the retailer's profit and integrates the S.C. Thus, the scheme can eliminate the effect of double-marginalisation. (5) As the supplier becomes more reliable, the retailer should increase the usage of PSPOF and purchasing quantity. [ABSTRACT FROM AUTHOR]

In this paper, we analyze some distributions involving the longest and shortest negative excursions of spectrally negative Lévy processes using the binomial expansion approach. More specifically, we study the distributions of such excursions and related quantities such as the joint distribution of the shortest and longest negative excursions and their difference (also known as the range) over a random and infinite horizon time. Our results are applied to address new Parisian ruin problems, stochastic ordering and the number near-maximum distress periods showing the superiority of the binomial expansion approach for such cases. [ABSTRACT FROM AUTHOR]

Often, reliability systems suffer shocks from external stress factors, stressing the system at random. These random shocks may have non ignorable effects on the reliability of the system. In this article, we provide sufficient (and necessary) conditions on components' lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two parallel systems with independent components by means of usual stochastic and reversed hazard rate orders, when the matrix of parameters changes to another matrix in a mathematical sense. We also discuss stochastic comparisons of series systems with independent components under random shocks in the sense of usual stochastic and hazard rate orders, by using the concept of vector majorization and related orders. The results developed in this article generalize some known results in the literature. [ABSTRACT FROM AUTHOR]

In the present work we first discuss the structure and properties of the ageing function of the Gumbel–Barnett copula in the exchangeable case. Then we define and illustrate the ageing notions such as IFR, DMRL, NBU etc for copulas using the weak copula ageing property of the Gumbel–Barnett copula. [ABSTRACT FROM AUTHOR]

ABSTRACT Multiple primary endpoints are commonly used in randomized controlled trials to assess treatment effects. When the endpoints are measured on different scales, the O'Brien rank‐sum test or the Wei–Lachin test for stochastic ordering may be used for hypothesis testing. However, the O'Brien–Wei–Lachin (OWL) approach is unable to handle missing data and adjust for baseline measurements. We present a nonparametric approach for data analysis that encompasses the OWL approach as a special case. Our approach is based on quantifying an endpoint‐specific treatment effect using the probability that a participant in the treatment group has a better score than (or a win over) a participant in the control group. The average of the endpoint‐specific win probabilities (WinPs), termed the global win probability (gWinP), is used to quantify the global treatment effect, with the null hypothesis gWinP = 0.50. Our approach involves converting the data for each endpoint to endpoint‐specific win fractions, and modeling the win fractions using multivariate linear mixed models to obtain estimates of the endpoint‐specific WinPs and the associated variance–covariance matrix. Focusing on confidence interval estimation for the gWinP, we derive sample size formulas for clinical trial design. Simulation results demonstrate that our approach performed well in terms of bias, interval coverage percentage, and assurance of achieving a pre‐specified precision for the gWinP. Illustrative code for implementing the methods using SAS PROC RANK and PROC MIXED is provided. [ABSTRACT FROM AUTHOR]

We investigate some information properties of consecutive k-out-of-n:G systems in light of fractional generalized cumulative residual entropy. We firstly derive a formula to compute fractional generalized cumulative residual entropy related to the system's lifetime and explore its preservation properties in terms of established stochastic orders. Additionally, we obtain useful bounds. To aid practical applications, we propose two nonparametric estimators for the fractional generalized cumulative residual entropy in these systems. The efficiency and performance of these estimators are illustrated using simulated and real datasets. [ABSTRACT FROM AUTHOR]

The objective in the study of reliability and survival theory is to have precision in the extracted information. The measure of dispersion in the information content has been newly introduced in the information theory, known as varentropy. Studying the variability in the uncertainty measure is found productive for stochastic systems conditional on survival. In this paper, we provide a novel stochastic order based on the notion of residual varentropy and study its properties. Further on the basis of the residual varentropy, novel classes of life distributions are identified and investigated in detail. Moreover, some applications of the proposed order are produced at the end. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the order estimation of an autoregressive model using the information criterion developed by El Matouat and Hallin (1996), which is based on stochastic complexity. This criterion is a generalization of the Hannan and Quinn criterion and provides a convergence of the model order estimator, but it depends on a parameter that is sensitive to the sample size. In order to select the exact order of the candidate model, we propose a method for identifying the values of this parameter from the sample using the information contained in sub-samples of increasing size. To study the performance of the proposed method in comparison with the usual criteria, we simulated samples from autoregressive models on which we applied our procedure. Simulation results support the relevance of our procedure when compared to the Akaike criterion, the Hannan and Quinn criterion, and the Schwarz criterion. [ABSTRACT FROM AUTHOR]

Stochastic analysis of the absorbed power density (APD) on skin surface exposed to radiation of halfwave dipole at 10, 30 and 90 GHz and for different antenna‐body distances is presented. Skin tissue is modeled as a half‐space consisting of one layer (skin) or three layers (skin, fat and muscle) whose permittivities and conductivities are uncertain. Deterministic part is based on numerical solution of Pocklington equation via Galerkin Bubnov Indirect Boundary Element method (GB‐IBEM) and numerical integration of the corresponding field integrals. Uncertainty from tissue electric parameters is propagated to APD via non‐intrusive Stochastic Collocation method (SCM) in order to compute stochastic moments of APD. For 1‐layered model APD stochastic moments are computed with 3 deterministic simulations. On the other hand APD mean and variance for 3 layered model are successfully computed with 13 deterministic simulations while skewness and kurtosis require 85 deterministic simulations. The ratio of APD standard deviation and mean decreases with frequency thus indicating that the uncertainty in the tissue electric properties has smaller effect on APD uncertainty at higher frequencies. Finally, sensitivity analysis carried out for both 1‐layered and 3‐layered models indicates the same conclusions. At 10 GHz skin permittivity and conductivity are the two most important parameters. However, as frequency increases the impact of skin conductivity prevails. This indicates that in frequency range 10–90 GHz the APD uncertainty quantification and sensitivity analysis can be carried out by using only 2‐dimensional stochastic model. Key Points: The reported values for tissue permittivity and conductivity are scattered, thus introducing the uncertainty in the output absorbed power density (APD)Stochastic Collocation can assure the computation of APD stochastic moments with good accuracy at relatively low number of simulationsSensitivity analysis shows that the impact of the inner layers' permittivity and conductivity is negligible [ABSTRACT FROM AUTHOR]

The article proposes a sequential statistical procedure to test for the presence of level shifts affecting bounded time series, regardless of their order of integration. The article shows that bounds are relevant for the statistic that assumes that the time series are integrated of order one. In contrast, they do not affect the limiting distribution of the statistic that is defined for time series that are integrated of order zero. The article proposes a union rejection statistic for bounded processes that does not require information about the order of integration of the stochastic processes. The model specification is general enough to consider the existence of structural breaks that can affect either the level of the time series and/or the bounds that limit its evolution. Monte Carlo simulations indicate that the procedure works well in finite samples. An empirical application that focuses on the Swiss franc against the euro exchange rate evolution illustrates the usefulness of the proposal. [ABSTRACT FROM AUTHOR]

In this paper, we focus on two relative aging classes, namely increasing (decreasing) relative failure rate and increasing (decreasing) failure rate relative to average failure rate. We studied some reliability properties and connections with other classes of lifetime distributions. The main objective of this paper was to investigate the preservation properties of decreasing relative failure rate class and decreasing failure rate relative to average failure rate class under the structure of ( n − k + 1)-out-of- n system. We give some examples of parametric distributions to evaluate the correctness of the results. [ABSTRACT FROM AUTHOR]

In analyzing bivariate data sets, data with common observations are frequently encountered and, in this case, existing absolutely continuous bivariate distributions are not applicable. Only a few models, such as the bivariate distribution proposed by Marshall and Olkin (J Am Stat Assoc 62(317):30–44, 1967), have been developed to model such data sets and the choice of models to fit data sets having common observations is very limited. In this paper, three general classes of bivariate distributions for modeling data with common observations are developed. To develop the bivariate distributions, we employ a probability model in reliability. Considering a system with two components, it is assumed that, when the first failure of the components occurs, with some probability, it immediately causes the failure of the remaining component, and, with complementary probability, the residual lifetime of the remaining component is shortened according to some stochastic order. It will be shown that, by specifying the underlying distributions contained in the joint distribution, numerous families of bivariate distributions can be generated. Therefore, this work provides substantially increased flexibility in modeling data sets with common observations. The developed models are fitted to two real-life data sets and it is shown that these models outperform the existing models in terms of fitting performance and their performances are satisfactory. [ABSTRACT FROM AUTHOR]

We study copula-based collective risk models when the dependence structure is defined by a Farlie-Gumbel-Morgenstern (FGM) copula. By leveraging a one-to-one correspondence between the class of FGM copulas and multivariate symmetric Bernoulli distributions, we find convenient representations for the moments and Laplace-Stieltjes transform for the aggregate random variable defined from collective risk models with FGM dependence. We examine different components of this collective risk model, aiming to better understand the impact of the assumed dependence between a claim's frequency and severity. Relying on stochastic ordering, we analyze the impact of dependence on the aggregate claim amount random variable. Even if the FGM copula may only induce moderate dependence, we illustrate through numerical examples that the cumulative effect of FGM dependence can lead to substantial variations in key risk measures on aggregate random variables defined from collective risk models. [ABSTRACT FROM AUTHOR]

This paper considers parallel and series systems with heterogeneous components having dependent exponential lifetimes. The underlying dependence is assumed to be Archimedean and the component lifetimes are supposed to be connected according to an Archimedean copula. Sufficient conditions are found to dominate a parallel system with heterogenous exponential components, with respect to the dispersive order, by another parallel system with homogenous exponential components where the dependence structure between lifetimes of components is the same. We also compare two series systems (and two parallel systems) with general one-parameter dependent components and with respect to the usual stochastic ordering. Examples are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper studies some stochastic properties of random quantiles according to the newly defined reliability measure called reversed aging intensity function. Preservation property of reversed aging intensity order under random quantile is obtained and using it, a lower bound and an upper bound for the reversed aging intensity function of a random quantile are derived. Preservation of two related monotonic reliability classes under random quantiles is also studied. We finally apply our results for reliability analysis of series systems with heterogeneous component lifetimes. Examples are included to examine and analyze the obtained results. [ABSTRACT FROM AUTHOR]

Stochastic ultimate boundedness has always been a very important property, which plays an important role in the study of stochastic models. Thus, in this paper, we will study a stochastic periodic chemostat system, in which we assume that the nutrient input concentration and noise intensities are periodic. In order to make the stochastic periodic model have mathematical and biological significance, we will study a very important issue: the existence, uniqueness and ultimate boundedness of a global positive solution for a stochastic periodic chemostat system. [ABSTRACT FROM AUTHOR]

Higher order risk measures are stochastic optimization problems by design, and for this reason they enjoy valuable properties in optimization under uncertainties. They nicely integrate with stochastic optimization problems, as has been observed by the intriguing concept of the risk quadrangles, for example. Stochastic dominance is a binary relation for random variables to compare random outcomes. It is demonstrated that the concepts of higher order risk measures and stochastic dominance are equivalent, they can be employed to characterize the other. The paper explores these relations and connects stochastic orders, higher order risk measures and the risk quadrangle. Expectiles are employed to exemplify the relations obtained. [ABSTRACT FROM AUTHOR]

The use of redundancies or spares in a system is a widely adopted technique to enhance system reliability and reduce the risk of system failure. Redundancies are typically incorporated into systems at the component or system levels. It is a significant problem to allocate appropriate redundancies into a system from a set of available options for the same. In this paper, we establish sufficient conditions to compare the reliability of coherent systems of dependent components with different sets of active redundancy, whether at the component level or the system level, based on some stochastic orders. We have obtained the results for the component lifetimes following the proportional odds (PO) model (the Marshall–Olkin family of distributions) for any lifetime distribution as a baseline distribution. We have studied the problem in the most general setup, with the consideration of coherent system that includes most of the common system structures, the consideration of non-matching spares, the consideration of dependencies of the components with different associated parameters of the copulas, and the consideration of general distribution as the baseline distribution of the PO model. We provide examples satisfying the sufficient conditions of the theoretical results. Additionally, we illustrate some of the results using real-world data. [ABSTRACT FROM AUTHOR]

In system reliability, coherent systems play a central role since without them any system is considered flawed. A system is coherent if all the components are relevant to that system i.e. functioning of every component has an effect in the functioning of the system and the structure of the system is monotone. Monotone structure means that improving any component will not deteriorate the system. A mixed system is a stochastic mixture of coherent systems and any coherent system is a special case of a mixed system. In reliability engineering, one major problem is to compare various coherent and mixed systems so that the better system can be used to increase the overall reliability. Another important problem is to measure the complexity of systems. A highly complex system will naturally have a higher running and maintenance cost associated with it and it is desirable for a reliability engineer to have an understanding regarding the complexity of systems beforehand. In this paper, we address these two problems from an information theoretic approach. Extropy is a measure of information which is the dual of the famous Shannon entropy measure. Recently, a new measure related to extropy, called cumulative residual extropy (CREx), was introduced in the literature by Jahanshahi et al. (Probab Eng Inf Sci 1–21, 2019). This measure is based on the survival function of the underlying random variable and it has some advantages over extropy measure. In this work, we analyze the CREx measure for coherent and mixed systems and develop some comparison results among systems. We also obtain some bounds of CREx of coherent and mixed systems consisting of independent and identically distributed (iid) and dependent and identically distributed (d.i.d.) components. We propose a new divergence measure to calculate the complexity of systems having iid components. Also, we introduced a new discrimination measure to compare various coherent and mixed systems when pairwise comparisons by usual stochastic order is not possible. Finally, we discuss analysis of the CREx measure of coherent systems having heterogeneous components. We also provide applications involving redundancy allocation. Various numerical examples are considered for illustrative purposes. [ABSTRACT FROM AUTHOR]

In this article, the generalized total time on test transform and some related transforms for weighted variables are stated. Their characteristics and relationship with each other have been considered and also these transforms have been investigated in the weighted mode from the point of view of stochastic orders. Also, by presenting graphs of generalized total time on test transform for some common weight functions, its behavior with respect to the weighted function is studied. Then the relationship of this transform with its initial state is expressed. In the following, the topic under discussion is explained with some practical examples. Then providing a comprehensive exploration of the applications of the studied transforms within the domains of insurance and reliability. By delving into these practical contexts, we gain valuable insights into how these mathematical tools can be effectively utilized to address complex challenges in risk assessment, decision-making, and resource allocation. Additionally, the examination of the NBU class of distributions offers a deeper understanding of their behavior, shedding light on their relevance and applicability in various statistical analyses. Finally, the article concludes with a detailed discussion of a specific real dataset, offering a concrete demonstration of how the topic under study can be applied in practice. [ABSTRACT FROM AUTHOR]

The article considers the multivariate stochastic orders of upper orthants, lower orthants and positive quadrant dependence (PQD) among simple max-stable distributions and their exponent measures. It is shown for each order that it holds for the max-stable distribution if and only if it holds for the corresponding exponent measure. The finding is non-trivial for upper orthants (and hence PQD order). From dimension d ≥ 3 these three orders are not equivalent and a variety of phenomena can occur. However, every simple max-stable distribution PQD-dominates the corresponding independent model and is PQD-dominated by the fully dependent model. Among parametric models the asymmetric Dirichlet family and the Hüsler-Reiß family turn out to be PQD-ordered according to the natural order within their parameter spaces. For the Hüsler-Reiß family this holds true even for the supermodular order. [ABSTRACT FROM AUTHOR]

Random weighted k$$ k $$‐out‐of‐n$$ n $$ systems are very useful in modeling the lifetimes of systems, wherein the success or failure of a system depends not only on its current operational status, but also on the contributions made by its components. In this paper, we consider random weighted k$$ k $$‐out‐of‐n$$ n $$ systems with redundant components drawn randomly from a mixed population consisting of m$$ m $$ different subpopulations/substocks. We study different optimal allocation policies of active redundancies and minimal repair components in a random weighted k$$ k $$‐out‐of‐n$$ n $$ system. Moreover, we investigate how the heterogeneity of subpopulations of items impacts the lifetime of a random weighted k$$ k $$‐out‐of‐n$$ n $$ system. We also present some simulational results and a real data analysis for illustrative purpose. [ABSTRACT FROM AUTHOR]

Motivated by recent works about stochastic comparisons between extreme order statistics arising from heterogeneous and dependent random variables, where the dependency structure is defined by the family of Archimedean copulas and the marginal distributions follow some specific parametric distributions, in this work, we investigate the case in which the marginal distributions can have arbitrary distribution functions depending on some parameter. Such parameter can be a shape, scale or location parameter, but other kinds of parameters, as frailty, resilience or tilt parameters can be also considered. Hence, the modified proportional hazard rate scale (MPHRS) and the modified proportional reversed hazard rate scale (MPRHRS) models, among others, belong to the wide parametric model studied here. Under this setup, we provide some general results for the usual stochastic order, when the parameter vectors verify the p-larger order or the reciprocally majorization order, generalizing some of the existing results in the literature. Besides this, extreme order statistics arising from the dependent MPHRS and MPRHRS models are compared in the sense of the reversed hazard rate order and the hazard rate order as well. [ABSTRACT FROM AUTHOR]

This paper explored the concept of past Rényi entropy within the context of k-record values. We began by introducing a representation of the past Rényi entropy for the n-th lower k-record values, sampled from any continuous distribution function F, concerning the past Rényi entropy of the n-th lower k-record values sampled from a uniform distribution. Then, we delved into the examination of the monotonicity properties of the past Rényi entropy of k-record values. Specifically, we focused on the aging properties of the component lifetimes and investigated how they impacted the monotonicity of the past Rényi entropy. Additionally, we derived an expression for the n-th lower k-records in terms of the past Rényi entropy, specifically when the first lower k-record was less than a specified threshold level, and then investigated several properties of the given formula. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of generalized conditional spacings and establishes partial order relations between different generalized spacings. First, we derive the survival function of the generalized conditional spacings. Second, we construct the stochastic and hazard rate order relationships between different generalized conditional spacings and generalized normal conditional spacings, considering parent distributions that belong to the decreasing failure rate (DFR) and increasing likelihood rate (ILR) classes. Finally, for parent distributions within the DFR class, we obtain the dispersive order between different conditional spacings, along with an inequality for the variance. Additionally, we present illustrative examples involving Pareto and Gamma distributions. [ABSTRACT FROM AUTHOR]

In this work, we establish some stochastic comparison results for multivariate skew-elliptical random vectors. These multivariate stochastic comparisons involve Hessian and increasing-Hessian orderings and many of their special cases. We provide necessary and/or sufficient conditions for the orderings by comparing the underlying model parameters. In addition, we investigate the (positive) linear forms of usual stochastic, convex and increasing convex, positive convex and increasing-positive-convex orderings. Using these theoretical results, we explore two applications. The first involves determining the upper bound of multivariate skew-elliptical risk variables under specific parameter constraints. The other one focuses on assessing the portfolio aggregation risks. Finally, two examples based on numerical simulations and real data from an Australian insurance company illustrate the established results and practical explanations. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of dynamic cumulative residual extropy inaccuracy (DCREI) by expanding on the existing dynamic cumulative residual extropy (DCRE) measure and proposes a weighted version of it. The paper then investigates a characterization problem for the proposed weighted dynamic extropy inaccuracy measure under the proportional hazard model and characterizes some well-known lifetime distributions using the weighted dynamic cumulative residual extropy inaccuracy (WDCREI) measure. Additionally, the study discusses the stochastic ordering of WDCREI and certain results based on it. Non-parametric estimations of the proposed measures based on kernel and empirical estimators are suggested. Results of a simulation study show that the kernel-based estimators perform better than the empirical-based estimator. Finally, applications of the proposed measures on model selection are provided. [ABSTRACT FROM AUTHOR]

Sequential order statistics (SOS) are useful tools for modeling the lifetimes of systems wherein the failure of a component has a significant impact on the lifetimes of the remaining surviving components. The SOS model is a general model that contains most of the existing models for ordered random variables. In this paper, we consider the SOS model with non-identical components and then discuss various univariate and multivariate stochastic comparison results in both one-and two-sample scenarios. [ABSTRACT FROM AUTHOR]

There are some connections between aging notions, stochastic orders, and expected utilities. It is known that the DRHR (decreasing reversed hazard rate) aging notion can be characterized via the comparative statics result of risk aversion, and that the location-independent riskier order preserves monotonicity between risk premium and the Arrow–Pratt measure of risk aversion, and that the dispersive order preserves this monotonicity for the larger class of increasing utilities. Here, the aging notions ILR (increasing likelihood ratio), IFR (increasing failure rate), IGLR (increasing generalized likelihood ratio), and IGFR (increasing generalized failure rate) are characterized in terms of expected utilities. Based on these observations, we recover the closure properties of ILR, IFR, and DRHR under convolution, and of IGLR and IGFR under product, and investigate the closure properties of the dispersive order, location-independent riskier order, excess wealth order, the total time on test transform order under convolution, and the star order under product. We have some new findings. [ABSTRACT FROM AUTHOR]

To investigate potentially dependent lifetimes, it is necessary to extend the $ \alpha $-quantile inactivity time to bivariate and multivariate frameworks. To extend this measure to a dynamic multivariate framework, all possible trajectories at time $ t $ are considered. The behavior of the extended $ \alpha $-quantile of inactivity time was investigated in relation to the corresponding multivariate hazard rate function. The $ \alpha $-quantile of the inactivity order is defined and discussed for the multivariate case. The difference between the two bivariate $ \alpha $-quantile functions of inactivity, which is an important measure for studying the effect of treatment on lifespan, was also investigated. This measure was used to analyze the effect of laser treatment on the delay of blindness. Two bootstrap approaches were implemented to construct confidence bounds for the difference measure.

Facing uncertain future customer orders, a pull-based available-to-promise (ATP) mechanism will deteriorate the overall profit since it allocates critical resources only to current customer orders. To prevent current less-profitable customer orders from over-consuming critical resources, this study investigates a push–pull based ATP problem with two time stages and three profit margin levels, and develops a dynamic resource reservation policy to maximise the expected total profit. Then, a corresponding push–pull based stochastic ATP model is established with known independent demand distributions, and the optimal reservation level is derived by the genetic algorithm to maximise the expected total profit. Finally, a series of simulation experiments are conducted to reveal the impact of some key factors, and the experiment results provide theoretical guidance and implementation methods for companies to maximise overall profits. [ABSTRACT FROM AUTHOR]

We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k -th power is also a trace-zero DS matrix for k ∈ { 2 , 3 , 4 , 5 }. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5. [ABSTRACT FROM AUTHOR]

In this study, we consider two finite mixture models (FMMs) with location-scale family distributed components, in which ordering results are established in various stochastic senses. For heterogeneity in one parameter, the comparisons are obtained with respect to usual stochastic order, hazard rate order, reversed hazard rate order and likelihood ratio order. Further, for heterogeneity in two parameters, we derive sufficient conditions for the stochastic comparison of FMMs with respect to usual stochastic order and hazard rate order. Various examples and counterexamples are presented to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$. With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as 'model aggregation'. [ABSTRACT FROM AUTHOR]

We show that for arrival processes, the 'harmonic new better than used in expectation' (HNBUE) (or 'harmonic new worse than used in expectation', HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities. [ABSTRACT FROM AUTHOR]

This paper explored the concept of past Rényi entropy within the context of $ k $-record values. We began by introducing a representation of the past Rényi entropy for the $ n $-th lower $ k $-record values, sampled from any continuous distribution function $ F, $ concerning the past Rényi entropy of the $ n $-th lower $ k $-record values sampled from a uniform distribution. Then, we delved into the examination of the monotonicity properties of the past Rényi entropy of $ k $-record values. Specifically, we focused on the aging properties of the component lifetimes and investigated how they impacted the monotonicity of the past Rényi entropy. Additionally, we derived an expression for the $ n $-th lower $ k $-records in terms of the past Rényi entropy, specifically when the first lower $ k $-record was less than a specified threshold level, and then investigated several properties of the given formula.

AbstractIn the context of actuarial science, the second largest claim amount is crucial to insurance analysis since they provide useful information for determining annual premium. In this article, we provide sufficient conditions of the second largest claim amounts arising from two sets of dependent and heterogeneous individual risk models according to various stochastic orders. It is first shown that the reversed hazard order between the occurrence probabilities vectors implies the reversed hazard order of the second largest claim amounts under certain conditions. Second, sufficient conditions are established for the usual stochastic ordering of the second largest claim amounts arising from heterogeneous dependent individual risk models under copula dependence. Finally, several examples illustrating the theoretical results are presented as well. [ABSTRACT FROM AUTHOR]

Abstract.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. [ABSTRACT FROM AUTHOR]

Related to a stochastic investment problem which aims to deter-mine when is it better to first invest a larger amount of money and afterwards a smaller one, in this paper we introduce a new preference relation between random variables. We investigate the link between this new relation and some well-known stochastic order relations and present some characterization properties illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

We consider here two finite (arithmetic) mixture models (FMMs) with general parametric family of distributions. Sufficient conditions for the usual stochastic order and hazard rate order are then established under the assumption that the model parameter vectors are connected in p-larger order, reciprocal majorization order and weak super/sub majorization order. Furthermore, we establish hazard rate order and reversed hazard rate order between two mixture random variables (MRVs) when a matrix of model parameters and mixing proportions changes to another matrix in some mathematical sense. We have also considered scale family of distributions to establish some sufficient conditions under which the MRVs have hazard rate order. Several examples are presented to illustrate and clarify all the results established here. [ABSTRACT FROM AUTHOR]

The Marshall-Olkin family of probability distributions has been the inspiration of numerous research publications in the field of probability distributions. In this paper, we present several new properties of this family. In particular, we focus on stochastic orders, stress-strength reliability, Lorenz and the Leimkhuler curves, compounding, and integrated tail distribution. Two applications related to Lorenz curves and ruin theory are finally presented. [ABSTRACT FROM AUTHOR]

The paper analyses optimal spending of an endowment fund. The purpose is to find a spending rule which is optimal for the owners and which secures that the fund will last "forever". This we do by finding closed form solutions of the optimal consumption to wealth ratio. We solve this problem using the life cycle model, where the agent can have preferences represented by expected utility or recursive utility. We apply our results to a sovereign wealth fund, and demonstrate that the optimal spending rate is significantly lower than the fund's expected real rate of return, a rule which is in common use. Employing the latter as the spending rate, implies that the fund's value deteriorates both in probability and in expectation, as time goes. For both kinds of long term convergence we find closed form threshold values. Spending below these values secures a sustainable fund. [ABSTRACT FROM AUTHOR]

The geometric process has been widely studied in various disciplines and applied in reliability, maintenance and warranty cost analysis, among others. In its applications in maintenance policy optimisation, the geometric process assumes constant repair effectiveness by its process rate. Nevertheless, in practice, maintenance effectiveness may differ from time to time and can therefore be better depicted by a random variable. Motivated by this argument, this paper proposes a new variant of the geometric process, which is referred to as the rate randomized geometric process (RRGP). The probabilistic properties of the RRGP are then investigated. The maximum likelihood method is utilised to estimate the parameters of the RRGP. Numerical examples are given to show its applicability in both maintenance policy optimization and fitting real‐world failure datasets. [ABSTRACT FROM AUTHOR]

Abstract.Frailty and resilience models provide a way to introduce random effects in hazard and reversed hazard rate modeling by random variables, called frailty and resilience random variables, respectively, to account for unobserved or unexplained heterogeneity among experimental units. This article investigates the effects of frailty and resilience random variables on the baseline random variables using some shifted stochastic orders based on some ageing properties of the baseline random variables. Relevant examples are presented to illustrate the findings, and the applicability of the results is demonstrated with real-world data, highlighting their relevance in practical scenarios. [ABSTRACT FROM AUTHOR]

This study compares two different series systems under various scenarios for which the components are assumed to be heterogeneous and follow Gompertz-G distributions. In the first scheme, the components of the systems are supposed to be independently distributed. In the second, we compared two series systems in the case that the independent components also experience random shocks. In the last scenario, we considered a case where the components of the systems have dependent structure sharing Archimedean copula. However, in all scenarios, the comparisons are performed based on the concepts of usual stochastic, the hazard rate, and the likelihood ratio orders through the majorization of the Gompertz-G parameters. [ABSTRACT FROM AUTHOR]

Systemic risk is the risk that a company- or industry-level risk could trigger a huge collapse of another or even the whole institution. Various systemic risk measures have been proposed in the literature to quantify the domino and (relative) spillover effects induced by systemic risks such as the well-known CoVaR, CoES, MES and CoD risk measures, and associated contribution measures. This paper proposes another new type of systemic risk measure, called the joint marginal expected shortfall (JMES), to measure whether the MES of one entity's risk-taking adds to another one or the overall risk conditioned on the event that the entity is already in some specified distress level. We further introduce two useful systemic risk contribution measures based on the difference function or relative ratio function of the JMES and the conventional ES, respectively. Some basic properties of these proposed measures are studied such as monotonicity, comonotonic additivity, non-identifiability and non-elicitability. For both risk measures and two different vectors of bivariate risks, we establish sufficient conditions imposed on copula structure, stress levels, and stochastic orders to compare these new measures. We further provide some numerical examples to illustrate our main findings. A real application in analyzing the risk contagion among several stock market indices is implemented to show the performances of our proposed measures compared with other commonly used measures including CoVaR, CoES, MES, and their associated contribution measures. [ABSTRACT FROM AUTHOR]

In the past few years, considerable attention has been given to the extropy measure. The extropy and weighted extropy of ranked set sampling were studied by several authors. The general weighted extropy and some results related to it are introduced in this paper. We provide general weighted extropy of ranked set sampling. We also studied characterization results, stochastic comparison and monotone properties of general weighted extropy. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for stochastic comparisons of two finite mixture models (FMMs) with respect to the usual stochastic order, hazard rate order, and likelihood ratio order when the mixing components have generalized Weibull family of distributions. The established (sufficient) conditions are mainly based on the majorization order, weak supermajorization order, and weak submajorization order. The stochastic comparisons are studied when there is heterogeneity in one (model) parameter, and then in two parameters (model parameter and mixing proportion). Further, the concept of unordered majorization order is employed to establish the usual stochastic order between two FMMs. To illustrate the theoretical results established here, several numerical examples and counterexamples are presented. Finally, we have generalized some of the results to the case of τ-mixture models. [ABSTRACT FROM AUTHOR]

Warehouses recently face increasing stress imposed by a volatile customer demand and increasing customer expectations in terms of ever shorter order response times. In that respect, warehouses more and more offer same-day and next-day shipment conditions. However, same-day shipment promises are challenging to fulfil, especially as the order fulfilment process operates against fixed deadlines imposed by the predefined truck departure times. As a natural mitigation strategy, warehouses set a cutoff point and offer same-day shipment only to customers that order until the cutoff point, but next-day shipment to all customers ordering thereafter. Setting an appropriate cutoff point is challenging as it affects multiple facets of the service quality, such as the order response time and the service level. In this paper, we study the design of cutoff-based shipment promises for stochastic deadline-oriented order fulfilment processes in warehouses. We present a discrete-time Markov chain model for exact steady-state performance analysis and propose two novel performance measures – α - and β - cutoff service level – for service level measurement in these systems. We numerically show the benefit of cutoff-based shipment promises. Even with a late cutoff point, there is a significant gain in the system performance. Furthermore, we find that warehouses should set the cutoff point such that it balances customer expectations in terms of service level and order response time. Finally, warehouses can improve their shipment promises when referring to β - instead of α - cutoff service level and by implementing measures reducing the utilisation and the variabilities of the order fulfilment process. [ABSTRACT FROM AUTHOR]

Redundancy allocations strategies are commonly used in reliability engineering to enhance the performance of reliability systems. This paper makes use the hierarchical copula to characterize the dependence structures among the components and hot/cold standbys in series systems. Optimal allocation policies of hot and cold standbys are presented for series systems with dependent components and redundancies under certain conditions imposed on the hierarchical copula as well as the components and spares reliabilities. It is shown that, under certain mild conditions imposed on the copula function and lifetimes of components and redundancies, the optimal allocations should be following the way that the resultant lifetimes for all nodes should be balanced as much as possible to enhance the system's performance. The results extend some related ones well‐known in the literature from the independent case to the dependent setting. Some numerical examples are presented as illustrations. We also present a real application in optimal allocations of redundant wires for improving the reliability of high‐voltage electricity transmission network systems. [ABSTRACT FROM AUTHOR]