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A new family of multivariate copulas constructed from existing copulas by transforming the corresponding random variables using their conditional distributions are introduced. It is shown that for symmetrical copulas in a triangular scheme, this transformation tends to asymptotic independence.

Let X k 1 k = 1 ∞ , X k 2 k = 1 ∞ , … , X k d k = 1 ∞ be d independent sequences of Bernoulli random variables with success-parameters p 1 , p 2 , … , p d respectively, where d ≥ 2 is a positive integer, and 0 p j 1 for all j = 1 , 2 , … , d . Let S j ( n ) = ∑ i = 1 n X i j = X 1 j + X 2 j + ⋯ + X n j , n = 1 , 2 , … . We declare a “rencontre” at time n , or, equivalently, say that n is a “rencontre time,” if S 1 ( n ) = S 2 ( n ) = ⋯ = S d ( n ) . We motivate and study the distribution of the first (provided it is finite) rencontre time.

Existing structures may suffer from resistance deterioration due to repeated attacks. The modeling of resistance deterioration is a critical ingredient in the reliability assessment and service life prediction of these degraded structures. In this paper, an explicit compound Poisson process-based model is developed to describe the shock deterioration of structural resistance, where the magnitude of each shock deterioration increment is modeled by a Gamma-distributed random variable. The moments (mean value and variance) and the distribution function of the cumulative shock deterioration are derived in a closed form, based on a proposed W-function. A method for the efficient calculation of the W-function is presented, which reduces to the Bessel type I function if the shock deterioration increment is exponentially distributed (a special case of Gamma distribution). The proposed shock deterioration model is applicable to either a stationary or a nonstationary Poisson process of random jumps. Subsequently, the overall resistance deterioration is modeled as the linear combination of gradual and shock deteriorations, based on which the proposed model can be used in the time-dependent reliability assessment of aging structures efficiently. A numerical example is presented to demonstrate the applicability of the proposed deterioration model by estimating the time-dependent reliability of an aging bridge. It is found that a smaller threshold for the degraded resistance leads to greater mean value and standard deviation of the time to failure, and this effect is enhanced by a smaller occurrence rate of the shock deterioration.

Versions of several results from the theory of random variables are proved for fuzzy random variables: the Skorokhod representation theorem, the Vitali convergence theorem, the dominated convergence theorem, the continuous mapping theorem, existence of regular conditional distributions, and a few others.

We consider an environment with m servers in parallel and n jobs. The n jobs have i.i.d. exponentially distributed processing requirements. The servers operate at different speeds and preemptions are allowed. The jobs have random release times that are not known in advance, but whenever a job is released, its due date is fixed and becomes known to the scheduler. We first consider three stochastic scheduling problems with three due date related objective functions and consider variations of the Earliest Due Date (EDD) rule, including the preemptive policy that at any point in time assigns the job with the Earliest Due Date to the Fastest Server (EDD-FS). Our optimality results turn out to be examples of stochastic scheduling problems that have relatively simple priority rules that are optimal while their deterministic counterparts do not allow such simple priority rules to be optimal. We furthermore extend our results to include a priority queueing model with m exponential servers that operate at different speeds and preemptions being allowed. The jobs arrive according to an arbitrary renewal process, and we assume that the utilization factor of the system is less than 1. Upon a job’s release the time difference between its due date and release time is set by the draw of a random variable from a given distribution. At a job’s release time, its due date is then immediately fixed and known while its actual processing time only becomes known upon its service completion. We show that for this priority queueing model the preemptive EDD-FS rule also minimizes the limit of several due date related objective functions as the number of jobs converges to infinity.

Let B n ( m ) be a set picked uniformly at random among all m-elements subsets of { 1 , 2 , … , n } . We provide a pathwise construction of the collection ( B n ( m ) ) 1 ⩽ m ⩽ n and prove that the logarithm of the least common multiple of the integers in ( B n ( ⌊ m t ⌋ ) ) t ⩾ 0 , properly centered and normalized, converges to a Brownian motion when both m , n tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of m independent random variables having uniform distribution on { 1 , 2 , … , n } . Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.

Empirical distributions in a range of fields are often substantially non-Gaussian but smooth enough to suggest that the underlying population distribution has at least several derivatives. Linear combinations of many random variables often have smooth densities even if the random variables are not independent. The main result here generalizes the elementary result that a convolution of densities has as many derivatives as the sum of the number of derivatives of each component to certain nonlinear functions of random vectors whose components may not be independent. Linearity is replaced by an assumption that the nonlinear function has positive first partial derivatives bounded away from 0 along at least some coordinates and independence is replaced by an assumption that the joint density has certain mixed partial derivatives. This approach to justifying the smoothness of empirical distributions is contrasted with other possible approaches, including solutions of stochastic partial differential equations and ergodic distributions for chaotic dynamical systems.

In this paper we consider a risk-averse newsvendor to assemble a final product that is made up of multiple complementary components, where both the product demand and the supply capacity of each component are random. A novel approach is proposed to derive the first-order condition for the optimal order quantity under the Conditional Value-at-Risk (CVaR) criterion. A comprehensive comparative statics analysis is conducted. The basic model is also extended to allow dependence among the random demand and supply capacities. We show that the objective function remains to be quasi-concave if those random variables are negatively dependent, and that the positive (negative) dependence of those random variables increases (decreases) the optimal order quantity and objective value. Numerical examples are provided to illustrate some of the main results.

In this paper, the robust filtering problem for uncertain complex networks with time-varying state delay and stochastic nonlinear coupling based on H ∞ performance criterion is studied. The random connections of coupling nodes are represented by utilizing independent random variables and the multiple fading measurements phenomenon is characterized by introducing diagonal matrices with independent stochastic elements. Moreover, the probabilistic time-varying delays in the measurement outputs are described by white sequences with the Bernoulli distributions. Furthermore, All system’s matrices are supposed to have uncertainty and a quadratic bound is assumed for nonlinear part of the network. This bound can be obtained by solving a sum of squares (SOS) optimization problem. By applying the Lyapunov theory, we design a robust filter for each node of the network so that the filtering error system is asymptomatically stable and the H ∞ performances are met. Then, the parameters of the filters are achieved by solving a linear matrix inequality (LMI) feasibility problem. Finally, the applicability and performance of the proposed H ∞ filtering approach are demonstrated via a practical example.

The omnipotent instrument for modeling multivariate dependence of random variables in standard probability theory has become copulas discovered by A. Sklar in 1959. Only recently Omladic and Stopar prove that in the bivariate case an analogous role is played by exactly the same copulas for random variables coming from finitely additive probability spaces. One of the main results of this paper is that this is true also in the general multivariate case. The extension to n dimensions requires a better understanding of quasi-copulas, the lattice closure of copulas with respect to the pointwise order. We need to develop a new equivalent definition of this notion that should be useful in other applications as well. Another tool we introduce and seems to be new even in the standard probability approach, is multivariate quasi-distributions. We also expand the coherence theory for quasi-copulas and quasi-distributions to the multivariate situation. Finally, our main result is a multivariate Sklar type theorem in the imprecise setting.

This paper considers the sliding mode control (SMC) problem of a class of uncertain Markovian jump systems, in which there exist randomly occurring parameter uncertainties and random gain variations in the controller. By introducing two independent random variables obeying Bernoulli distribution, the random characteristics of parameter uncertainties and controller gain variations are described. A mode-dependent sliding surface is constructed, and then, the non-fragile SMC scheme is synthesized such that the specified sliding surface is reached in finite time. Furthermore, the stochastic finite-time boundedness over both the reaching and sliding stages are ensured simultaneously under some sufficient conditions. Finally, the developed non-fragile SMC approach is verified by a practical example.

Level ice acting on vertical structures can impose large loads and cause damage. Phenomenological models have been developed to simulate ice-structure interaction and predict ice loads, however due to oversimplification of ice properties and interaction mechanisms, existing models either cannot predict the interaction well or are applicable to limited situations, such as when ice is brittle. This paper first presents observations from existing experiments to identify key features and underlying mechanisms. Using these insights, a new phenomenological model is developed that captures the important phenomena under different strain rates. The proposed model comprises three essential components. The first is a strain rate-dependent stress-strain relationship that controls ice failure and the failure length. A new equation is proposed for the stress-strain relationship, and the empirical parameters are calibrated from published compression tests. The second component entails the modeling of ice failure length as a random variable. The third component accounts for spatial correlation of the failure lengths, by dividing the ice edge into multiple zones, and imposing correlation between zones. The model is validated against rigid indentation tests, and the predictions are found to match the experimental data well in terms of effective pressure, peak pressure, and time series of ice load. By artificially removing each component and comparing the simulation result with experimental data, it is demonstrated that all three components are crucial for good performance of the proposed method.

A novel validation frame for deterministic solar irradiance forecasts is presented. It bases on the perception that a perfect forecast of a random variable must be indistinguishable from its observation. In mathematical terms: The random variables observation and forecast must be exchangeable. This implies that they have the same probability distribution and that the pertaining copula is symmetric. The validation frame tallies the equality of the probability distributions of observed and forecast solar irradiance, the symmetry of the pertaining copula, and the forecasting skill with well-defined measures, namely the Kolmogorov-Smirnov test statistic, copula asymmetry, and Spearman's ρ. The three measures are scale-invariant under strictly increasing transformations of observation and forecast. Thus, the frame is particularly suited for benchmarking solar irradiance forecasts from different locations, different calendar days, and over different periods. The frame is demonstrated on 24-h-ahead solar irradiance forecasts from two sites with different climates. Yet, it can be applied to any forecast of a continuous random variable. Additional theoretical insights connect the method to present-day practice, such as the Murphy-Winkler framework and the skill score.

As an efficient and robust tool in handling uncertainties, the first-order reliability method (FORM) is proposed for engineering problems under epistemic uncertainty. The method, namely dynamical relaxed directional method (DRD) coupled with an improved cuckoo search algorithm (ICOA) as DRD-ICOA consists of two numerical local and global analyzers, in which local analyzer performs reliability estimation to appraise the reliability index using the DRD approach. Whiles the global analyzer carries out the epistemic uncertainty of random variables using the alpha level set optimization by applying an ICOA. The DRD is formulated using the relaxed approach with a descent search direction by the dynamical sensitivity vector. The dynamical sensitivity vector is formulated using the sufficient decent conditions to guarantee the stability of FORM formula in structural fuzzy reliability analysis. Applicability of proposed numerical method of DRD-ICOA was examined via four engineering examples and was compared with several reliability methods-based FORM methods. Investigation results showed that the DRD Form could robustly converge to exact solutions and deal with real engineering problems via low computational burden.

We examine the influence of using a restart mechanism on the stationary distributions of a particular class of Markov chains. Namely, we consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We show that the restart mechanism may change significantly the limiting distribution. We obtain a limit theorem with a novel type of limiting distribution, a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.

An efficient and accurate reliability-based design of the geosynthetic-reinforced slopes (GRS) using the pseudo-static and modified pseudo-dynamic framework is proposed in the present study. Deterministic formulation used in the present study is made robust with the help of nonlinear constrained optimization. The collocation based stochastic response surface method (CSRSM) is used to probabilistically analyze the GRS. The critical modes of failure pertaining to the internal and external stability of the GRS are considered in the formation of the performance functions. The horizontal seismic acceleration coefficient (kh), internal friction angle of soil (φ), soil unit weight (γ), shear wave velocity (Vs), and friction angle at the interface between soil and reinforcement (φb) are chosen as the random variables, owing to their high influence on the stability of the GRS. The influence of correlation on the stability of the reinforced slope is illustrated considering the internal and external stability. System reliability analysis considering the internal and external modes of failure is also performed. An illustrative example is presented showing the steps to design a GRS using the proposed formulation. The results confirm the necessity of performing the system reliability analysis to estimate an accurate value of probability of failure of GRS.

The DEA is a nonparametric method of assessing the efficiency of decision-making units using mathematical programming. The classic DEA model assumes that input and output variables are deterministic. However, there are many applications where the variables are of a stochastic nature. Based on the consideration of input and output levels as random variables, the Stochastic Data Envelopment Analysis (SDEA) was developed. Statistical distributions therefore play a major role in this regard. By considering the dependency between input and output variables, and also their simultaneous dependencies in this study, we have introduced three copula-SCCR models. We used three copulas of Gaussian, Clayton, and Gumbel to estimate the dependence between the random variables with normal distribution. We evaluated the proposed models using real data from 20 bank branches. The results showed that considering stochastic dependency between the inputs or outputs causes different results. A comparison between the Copula-SCCR models and the SCCR models revealed that the efficiency of the DMUs using the Copula-SCCR models differed from the SCCR model by a significant margin of at least 20%.

We consider the model Y = X + ξ where Y is observable, ξ is a noise random variable with density f ξ , X has an unknown mixed density such that P ( X = X c ) = 1 − p , P ( X = a ) = p with X c being continuous and p ∈ ( 0 , 1 ) , a ∈ R . Typically, in the last decade, the model has been widely considered in a number of papers for the case of fully known quantities a , f ξ . In this paper, we relax the assumptions and consider the parametric error ξ ∼ σ N ( 0 , 1 ) with an unknown σ > 0 . From i.i.d. copies Y 1 , … , Y m of Y we will estimate ( σ , p , a , f X c ) where f X c is the density of X c . We also find the lower bound of convergence rate and verify the minimax property of established estimators.

This paper analyses the theory of the optimal output decision for a firm whose policy is to post a non-negotiable price for a good or service in a concentrated market where the demand facing the firm is determined, in part, by a random variable. The theoretical findings are the opposite of those in competitive markets; Proposition 1 states that the optimal output of a risk-averse firm is expected to be larger than that of a risk-neutral firm if the expected payoff of its marginal profit is less than or equal to 1. Proposition 2 states that the optimal output of a risk-seeking firm is expected to be smaller than that of a risk-neutral firm if the expected payoff of its marginal profit is greater than 1.

The limited sample size in multi-stress accelerated life testing makes the conventional large-sample-based inference methods inefficient. To overcome this problem, this paper develops a new statistical inference method based on random variable transformation for multi-stress accelerated life testing with Weibull distribution and progressive Type-II censoring. Firstly, a χ2 statistic is constructed using the random variable transformation method, and the exact point estimates of model parameters are derived based on the χ2 statistic. Subsequently, based on the χ2 statistic, the exact confidence interval of shape parameter is provided, while the generalized confidence intervals of the accelerated model parameters are calculated by constructing the new multivariate generalized pivotal quantities. Furthermore, the generalized confidence intervals of some commonly used reliability indexes are provided for better reliability management and decision making. Finally, a simulation study and a real case study are conducted to illustrate the implementation and effectiveness of the proposed method.

A group-characterizable (GC) random variable is induced by a finite group, called main group, and a collection of its subgroups. The notion extends directly to secret sharing schemes (SSSs). It is known that linear and abelian SSSs can be equivalently described in terms of GC SSSs. In this paper, we present a necessary and sufficient condition for a SSS to be equivalent to a GC one. Using this result, we show that homomorphic SSSs (HSSSs) are equivalent to GC SSSs whose subgroups are normal in the main group. We also present two applications for this equivalent description of HSSSs. One concerns lower bounding the information ratio of access structures for the class of HSSSs, and the other is about the coincidence between statistical, almost-perfect and perfect security notions for the same class.

This paper presents a stochastic formulation for capacitated single allocation hub location problems with uncertain demands, in which the balanced utilization of hub capacities is considered in the strategic decision making process. The demands are assumed to be independent random variables with known normal probability distributions. A stochastic programming model with joint chance constraints is established and then transformed into a second-order mixed-integer cone programming model. Furthermore, the proposed model is approximated by using piecewise tangent approximation and piecewise linear approximation techniques. For the approximated models, alternative reformulations are developed, and valid inequalities are employed to add to alternative reformulations. Extensive numerical experiments with CAB and AP data sets are conducted to evaluate the performance of the proposed methods, and analyze the configuration of hub-and-spoke networks and the utilization of hub capacities. Experimental results show that the optimal solution of proposed models can be obtained by using the two approximation techniques with a small number of tangent and linear segments. The developed alternative reformulations and valid inequalities can significantly improve computational efficiency. The entire unbalanced utilization degree of hub capacities can be greatly reduced with a small rise in the traditional operating cost.

We study the variance of the random variable that counts the number of lattice points in some shells generated by a special class of finite type domains in R d . The proof relies on estimates of the Fourier transform of indicator functions of convex domains.

We introduce unprecedented tools to infer approximate evolution features of the COVID19 outbreak when these features are altered by containment measures. In this framework we present: (1) a basic tool to deal with samples that are both truncated and non independently drawn, and (2) a two-phase random variable to capture a game changer along a process evolution. To overcome these challenges we lie in an intermediate domain between probability models and fuzzy sets, still maintaining probabilistic features of the employed statistics as the reference KPI of the tools. This research uses as a benchmark the daily cumulative death numbers of COVID19 in two countries, with no any ancillary data . Numerical results show: (i) the model capability of capturing the inflection point and forecasting the end-of-infection time and related outbreak size, and (ii) the out-performance of the model inference method according to conventional indicators.

For two independent Levy processes ξ and η and an exponentially distributed random variable τ with parameter q > 0 , independent of ξ and η , the killed exponential functional is given by V q , ξ , η ≔ ∫ 0 τ e − ξ s − d η s . Interpreting the case q = 0 as τ = ∞ , the random variable V q , ξ , η is a natural generalisation of the exponential functional ∫ 0 ∞ e − ξ s − d η s , the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional V q , ξ , η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫ 0 t e − ξ s − d η s for fixed t ≥ 0 , as well as for integrals of the form ∫ 0 ∞ f ( s ) d η s for deterministic functions f . Furthermore, applying the same techniques to the case q = 0 , new results on the absolute continuity of the improper integral ∫ 0 ∞ e − ξ s − d η s are derived.

This paper is concerned with the event-based fusion estimation problem for a class of multi-rate systems (MRSs) subject to sensor degradations. The MRSs under consideration include several sensor nodes with different sampling rates. To facilitate the filter design, the MRSs are transformed into a single-rate system (SRS) by using an augmentation approach. A set of random variables obeying known probability distributions are used to characterize the phenomenon of the sensor degradations. For the purpose of saving the limited communication resources, the event-triggering mechanism (ETM) is adopted to regulate the transmission frequency of the measurements. For the addressed MRSs, we aim to design a set of event-based local filters for each sensor node such that the upper bound of each local filtering error covariance (FEC) is guaranteed and minimized by designing the filter parameter appropriately. Subsequently, the local estimates are fused with the aid of covariance intersection (CI) fusion approach. Finally, a numerical experiment is exploited to demonstrate the usefulness of the developed fusion estimation algorithm.

We study a solution approach for a staffing problem in multi-skill call centers. The objective is to find a minimal-cost staffing solution while meeting a target level for the quality of service to customers. We consider a common situation in which the arrival rates are unobserved random variables for which preliminary forecasts are available in a first stage when making the initial staffing decision. In a second stage, more accurate forecasts are obtained and the staffing may have to be modified at a cost, to meet the constraints. This leads to a challenging two-stage stochastic optimization problem in which the quantities involved in the (nonlinear) constraints can only be estimated via simulation, so several independent simulations are required for each first-level scenario. We propose a solution approach that combines sample average approximation with a decomposition method. We provide numerical illustrations to show the practical efficiency of our approach. The proposed method could be adapted to several other staffing problems with uncertain demand, e.g., in retail stores, restaurants, healthcare facilities, and other types of service systems.

This article is devoted to the H ∞ estimation problem for stochastic semi-Markovian switching complex-valued neural networks subject to incomplete measurement outputs, where the time-varying delay also depends on another semi-Markov process. A sequence of random variables with known statistical property is introduced to depict the missing measurement phenomenon. Based on the generalized It o ˆ ’s formula in complex form concerning with the semi-Markovian systems, complex-valued reciprocal convex inequality as well as intensive stochastic analysis method, some mode-dependent sufficient conditions are presented guaranteeing the estimation error system to be exponentially mean-square stable with a prespecified H ∞ disturbance attenuation level. In addition, the mode-dependent estimator gain matrices are appropriately designed according to the feasible solutions of certain complex matrix inequalities. In the end, one numerical example is provided to illustrate effectiveness of the theoretical results.

We consider arbitrary subsets L of random variables defined on an arbitrary non-additive probability space ( Ω , F , ν ) . A topology τ on L satisfies Condition BU if every open set in this topology which contains X ∈ L as a member also contains as a subset some ( c , ϵ ) –ball around X, defined as B c , ϵ ( X ) = { Y ∈ L | ν ( | X − Y | ≥ c ) ϵ } . Condition BU is satisfied by any topology of convergence in non-additive measure ν [21] , [18] but also by all coarser topologies. Next we consider preference relations that are continuous with respect to any topology τ satisfying Condition BU. For non-atomic ν we prove that any convex and τ-continuous preference relation over the random variables in a given local cone L must satisfy monotonicity in the local cone order. This monotonicity result comes with surprisingly strong decision theoretic implications: (i) The only τ-continuous and convex preference relation defined over all random variables is the indifference relation; (ii) Any τ-continuous and convex preference relation defined over all positive random variables must satisfy payoff-monotonicity; (iii) Any convex and payoff-monotone preference relation defined over all loss random variables must violate τ-continuity.

This paper studies the problem of event-triggered control for the class of Markovian jump neural networks (MJNNs) under actuator saturation and hybrid cyber attacks. In order to save the limited network bandwidth, the event-triggered mechanism (ETM) is introduce to determine whether the signal of sampler is transmitted to the remote controller through the communication network. With the aid of two sets of Bernoulli distributed random variables (BDRVs), the mathematical model of randomly occurring deception attacks (RODAs) is presented. Due to the limitations of security and technology factors and the complex network environment in practice, actuator saturation and denial-of-service (DoS) attack are also considered. In summary, the MJNNs, ETM, actuator saturation and hybrid cyber attacks are incorporated into a unified construction, and a augmented system under this construction is modeled for the first time. For this system, the existence conditions of event-triggered control are derived through LyapunovKrasovskii functional (LKF). Based on this sufficient condition, the linear matrix inequality (LMI) technique is utilized to obtain the control gain of the controller and the weight matrix of the trigger. Finally, a numerical example is given to verify the effectiveness of the proposed method in this paper.

This article proposes a series of Bayesian learning and pricing models to uncover the unknown demand parameter through a data driven process and in the meantime to achieve a good regret bound for revenue. Treating the unknown demand sensitivity as a multinomial random variable whose parameters follow a Dirichlet prior, we take an advantage of conjugate property and derive the analytical form of posterior parameters. We show that the proposed learning process ensures the asymptotic convergence of the parameters. In addition to exploration, with few restrictive assumptions we develop a heuristic algorithm for revenue exploitation and establish a regret bound that is comparable to the best bounds available in the literature. We also show that if the estimated parameter falls into the neighborhood of its true value with sufficient learning, the heuristic can switch to the dynamic programming process. The revenue gap between the optimal dynamic programming process and its approximate counterpart becomes negligible. Numerical experiments confirm the theoretical conclusion. The heuristic algorithms, including a simplified two-price policy and an approximate dynamic programming heuristic, generally lead to an expected revenue with a small gap from its optimal level.

Let ( U , d ) be a complete separable metric space and ( F n ) n ≥ 0 a sequence of random functions from U to U . Motivated by studying the stability property for Markovian dynamic models, in this paper, we assume that the random function ( F n ) n ≥ 0 is driven by a Markov chain X = { X n , n ≥ 0 } . Under some regularity conditions on the driving Markov chain and the mean contraction assumption, we show that the forward iterations M n u = F n ∘ ⋯ ∘ F 1 ( u ) , n ≥ 0 , converge weakly to a unique stationary distribution Π for each u ∈ U , where ∘ denotes composition of two maps. The associated backward iterations M n u = F 1 ∘ ⋯ ∘ F n ( u ) are almost surely convergent to a random variable M ∞ which does not depend on u and has distribution Π . Moreover, under suitable moment conditions, we provide estimates and rate of convergence for d ( M ∞ , M n u ) and d ( M n u , M n v ) , u , v ∈ U . The results are applied to the examples that have been discussed in the literature, including random coefficient autoregression models and recurrent neural network.

Investigating the effect of measurement errors on the control chart monitoring the ratio of two normal random variables is an important task to facilitate the use of this kind of control chart in practice. Moreover, a deep insight into the problem can help practitioners to find a way to reduce unexpected impacts of measurement errors on the chart performance. This paper provides a study on the performance of the exponentially weighted moving average control chart monitoring the ratio in the presence of measurement errors. We extend the linear covariate error model applied in previous studies to a more general situation, which makes the study more realistic. The numerical results show that although the precision error and the accuracy error have negative influences on the proposed chart performance when these errors are not large these influences are not significant.

Numerous estimates exist of the factors appearing in the Drake equation, which estimates the number of extant communicative civilizations in the Milky Way galaxy. Although each are presented by scholars in the field, wide disagreement of the values of those factors exist. Here we consider the six non-longevity factors as random variables and utilize estimates from six studies to determine probability density functions for those factors. We then utilize them to perform a Monte Carlo analysis to determine the most likely value for the product. We find, in moderately close agreement with the estimate by both Frank Drake and the Cyclops report, a value for the product of 0.85 ± 1.28 years−1.

In this paper, a multi-objective portfolio optimization problem is studied in an uncertain random environment using higher moments. We consider a scenario involving an asset universe wherein some assets have sufficient historical return data for modelling as random variables, and others, listed relatively recently, lack historical data. The asset returns of such assets are modelled as uncertain variables. Thus, a hybrid environment involving both uncertain and random variables is considered. We use mean absolute semi-deviation as a risk measure and employ skewness (i.e., third moment) in the portfolio optimization model. The expressions for the mean absolute semi-deviation and skewness of an uncertain random variable have been derived. We show that the derivation of mean absolute semi-deviation is not based on any stipulation and thus is an accurate measure of risk (unlike the variance of an uncertain random variable, which uses stipulation). We propose a hybrid genetic algorithm as a solution methodology and provide empirical proofs to illustrate its advantages. The proposed methodology has been applied to a case study involving 100 assets listed in the NASDAQ-100 index of the U.S. stock market. We do an ex-post analysis to track the performance of our model out-of-sample and illustrate its advantages.

New in this work is a new analytical derivation of a Delay-Doppler map model for Fully-Focused SAR, which includes the vertical wave particle motion and a more general surface representation. It was indeed already noticed in Egido and Smith (2017) that the open ocean surfaces decorrelate within the integration time interval, which is used in the FF-SAR processing to form the synthetic antenna. However, this was not further discussed. The impact of surface motion is expected to be larger on FF-SAR than for unfocused SAR, since a longer integration time is used in the first processing. We introduce vertical wave particle velocities and accelerations as random variables in the analytical formulation, and relate this variables to ocean-wave spectral parameters, such as the mean zero up-crossing and crest period. Further on, we show and discuss for the first time that the flat sea surface response and the probability density function of the height of scattering surface elements are not the same as in unfocused SAR. Moreover, the response of a point target differs significantly and depends on the location of a scattering surface element, the satellite orbit parameters and the tracking range. This work provides the analytical expressions and a thorough discussion of these issues, including numerical results using CryoSat-2 mission parameters.

A novel fourth-order linear moment (L-moment) reliability method considering linear correlated (L-correlated) random variables is developed to improve accuracy and applicability on the premise of ensuring FORM's computational efficiency. For accomplishing this task, a novel normal transformation's implementation based on L-moments and L-correlations is established, and the complete monotonic expressions of Pearson correlation coefficient is obtained. Additionally, to guarantee that the specified L-correlation coefficient is valid, the suitable regions of L-correlation coefficient are identified based on the discriminant analysis. Then, the normal transformations are integrated in the FORM to evaluate the reliability of mechanical system with L-correlated variables. Several numerical examples are presented to demonstrate the accuracy, efficiency and applicability of the proposed method. Compared with FORM, the proposed method is improved by taking advantage of the fourth-order L-moments.

We consider random graphs G built on a homogeneous Poisson point process on R d , d ≥ 2 , with points x marked by i.i.d. random variables E x . Fixed a symmetric function h ( ⋅ , ⋅ ) , the vertexes of G are given by points of the Poisson point process, while the edges are given by pairs { x , y } with x ≠ y and | x − y | ≤ h ( E x , E y ) . We call G Poisson h -generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h ( a , b ) ≔ a + b and E x ≥ 0 . Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size n is lower bounded by C n d − 1 apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to h ( a , b ) = ( a + b ) γ with γ > 0 , the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller–Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.

The dynamic event-based asynchronous and resilient dissipative filter design for Markov jump singularly perturbed systems (MJSPSs) against stochastic deception attacks is discussed in this paper. Firstly, a novel dynamic event-based transmission protocol is provided to further decrease the proportion of sampled data into network. The effect of deception attacks is formulated as a random variable satisfying the Bernoulli distribution. And an asynchronous filter is delicately constructed. Based on the technique of linear matrix inequality (LMI), efficient criteria of stochastically stable for the filtering error systems with a predetermined dissipative performance are obtained. An effective method of jointly design the proposed dynamic event-triggered transmission protocol and the non-synchronous filter is offered. Lastly, a numerical instance and a resistance-capacitance (RC) circuit system are provided to display the effectiveness and the benefit of the developed method.

Maximum entropy (MAXENT) is a widespread method for recovering unknown probabilities of a random variable Z . The method uses first few (empiric) moments of Z and leads to non-parametric estimators. Here we study MAXENT in a Bayesian set-up assuming that there exists a well-defined Dirichlet density for unknown probabilities. This allows to employ the average Kullback-Leibler (KL) distance in evaluating various MAXENT constraints, and comparing MAXENT with parametric estimators: the regularized maximum likelihood (ML) and the Bayesian estimator. KL distance is singled out among other distances from demanding a weak consistency of MAXENT with respect to a shrinkage. We show that MAXENT applies to sparse data, provided that there are certain types of prior information, e.g. the probabilities of Z are most probably deterministic, or there exist prior conditional rank correlations between Z and its probabilities. In the latter case MAXENT can outperform the optimally regularized ML and produce results that are close to the globally optimal Bayes estimator. Predictions of MAXENT are meaningless (i.e. worse than a random guess) if the needed prior information is absent.

This study introduces a novel probability concept, random probability density function (PDF), as an efficient alternative of the random sampling for probability/reliability analysis of multivariate problems. To this end, a solution is proposed for drawing 1-D random PDFs according to the joint PDF of random variables. It is shown that using this approach, few random PDFs could be used instead of millions of random samples for covering the sample space in probabilistic analysis. By drawing PDFs instead of samples, the expected value of multivariable functions can be represented as the expected value of several one-variable probability integrals and therefore, any capable 1-D integration technique can be henceforth used for accurate statistical moment estimation and reliability analysis of multivariable problems. The law of large numbers proves the correctness of the proposed approach and therefore, achieving correct solutions of complex probability and reliability problems will not be hereafter restricted to only using crude Monte Carlo simulation. Besides, a novel probability density simulator is proposed that can attain probability density and moments of multivariate functions in an efficient manner different from existing approaches. Accurate and efficient probability/reliability analyses of complex problems with small failure probabilities confirm that the proposed method can be a reliable and efficient alternative of the crude Monte Carlo method, especially for extremely rare-events and time-consuming problems. For such problems, the suggested approach can serve as a benchmark against which new methods can be compared.

We study bilevel stochastic linear programs with fixed lower level feasible set and random follower's goal function and show that the parametrized random variable arising as the upper level outcome depends continuously on the leader's decision w.r.t. any L p -norm with p ∈ [ 1 , ∞ ) . This entails continuity of the objective function for a class of models involving convex risk measures defined on appropriate L p -spaces and allows to formulate verifiable sufficient conditions for the existence of optimal solutions.

Sensitivity analysis (SA) has been widely used in engineering analyses and designs because it provides information about the importance of input variables to a model response. In practical engineering problems, random and interval variables often coexist. Although some existing SA methods are applicable for the mixture of random and interval variables, most of them do not deal with the epistemic uncertainty of the model response and cannot be performed over a partial range of the response distribution, making it difficult to assist engineers effectively reduce the uncertain scope of the model response. In this paper, a new SA method that evaluates the impacts of parameters of input variables to the entire range or a partial range of the probability box (p-box) is developed. First, the area between the lower and upper bounds of the entire range or a partial range of the p-box is used to measure the uncertain scope of the model response. Three global sensitivity indices are then defined for the sensitivity of the area with respect to the parameters of interval and random variables. To efficiently and accurately calculate these indices, a predictive model based uncertainty analysis method is developed to determine the most probable points and values of the interval variables. The effectiveness of the proposed method is verified by two engineering examples.

Traditional multiple-discrete continuous choice models that have been formulated and applied in recent years consider a single linear resource constraint, which, when combined with consumer preferences, determines the optimal consumption point. However, in reality, consumers may face multiple resource constraints, such as those associated with time, money, and storage capacity. Ignoring such multiple constraints and instead using a single constraint can, and in general will, lead to poor data fit and inconsistent preference estimation, which can then have a serious negative downstream effect on forecasting and welfare/policy analysis. Unlike earlier attempts to address this multiple constraint situation, we formulate a new multiple-constraint (MC) multiple discrete continuous extreme value (MDCEV) model (or the MC-MDCEV model) that retains a closed-form probability structure and is as simple to estimate as the MDCEV model with one constraint. We achieve this by assuming a type-I extreme value distribution for the error term in its minimization form in the baseline utility preference of each good rather than a maximization form as in Bhat's (2005; 2008) original MDCEV formulation. The statistical foundation of the proposed model is based on the fact that the difference between a minimal type-I extreme value random variable with scale σ and the weighted sum of the exponential of standardized minimal type-I extreme value random variables (scaled up by σ) leads to an apparently new multivariate distribution that has an elegant and closed-form survival distribution function. Results from a simulation experiment show that our proposed model substantially outperforms single-constraint models; the results also highlight the serious mis-estimation that is likely to occur if only a subset of active constraints is used. The proposed model is applied to a case of week-long activity participation where individuals are assumed to maximize their utility from time-use subject to time and money budgets.It is hoped that our proposed simple closed-form multi-constraint MDCEV model will contribute to a new direction of application possibilities and to new research into situations where consumers face multiple constraints within a multiple discrete-continuous choice context.

We discuss when law-invariant convex functionals “collapse to the mean”. More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures. © 2021 The Author(s)

Randomness and fuzziness are two fundamental kinds of uncertain phenomena, which can be handled respectively by probability theory (sigma system) and possibility theory (max system). For many practical problems of information processing, random uncertainty may often co-exist with fuzzy uncertainty, which reminds us to look for a novel mechanism called sigma-max hybrid uncertainty inference. This mechanism can cope with randomness and fuzziness jointly and achieve a direct fusion of heterogeneous information modeled by probability or possibility. Specifically, we are interested in two typical forms of uncertainty inference, i.e., uncertainty update equation combining heterogeneous information as well as composition rule of heterogeneous relations. Such an objective is achieved by the adoption of joint description of a random variable and a fuzzy variable, i.e., hybrid distribution of probability and possibility, which lays the foundation for connecting the two uncertainty systems of probability and possibility, and results in the integrated sigma-max system. The hybrid distribution, other than joint probability (or possibility) distribution, has a couple of special normalization requirements that are order-dependent with respective to “addition” operation over random variable and “max’ operation over fuzzy variable. The derived sigma-max inference is applied to target recognition, an example of which is simulated to demonstrate the merit of the proposed method over the classic Bayesian classifier when randomness and fuzziness are both involved.

In this paper, the H ∞ sliding mode control (SMC) problem is discussed for a class of uncertain discrete networked systems with channel fading and randomly varying nonlinearities (RVNs) by employing the observer-based approach. Here, the Jth-order Rice fading model is adopted to describe the phenomenon of channel fading, where the channel coefficients are mutually independent random variables that take the values on certain interval. In addition, a sequence of mutually independent Bernoulli distributed random variables is utilized to characterize the phenomenon of the RVNs, where the occurrence probabilities of RVNs consist of the nominal mean and the bound of error. The main objective of the addressed problem is to design the observer-based sliding mode controller such that the closed-loop system is asymptotically stable in mean-square sense and the pre-specified H ∞ performance requirement is guaranteed. By employing novel Lyapunov–Krasovskii functional, sufficient criteria are derived to force the system state trajectories onto a pre-specified sliding mode region. Afterwards, the solvability of non-convex problem is discussed by testifying the feasibility of a minimization problem. Finally, an illustrative example is given to demonstrate the effectiveness of aforementioned SMC technique.

To characterize the regularity of distribution-path dependent SDEs in the initial distribution which varies as probability measure on the path space, we introduce the intrinsic and Lions derivatives for probability measures on Banach spaces, and prove the chain rule of the Lions derivative for the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is invalid, we establish the asymptotic Bismut formula. Both non-degenerate and degenerate noises are considered. The main results of this paper generalize and improve the corresponding ones derived recently in the literature for the classical SDEs with memory and McKean-Vlasov SDEs without memory.

The main goal of this paper is to study a new dispersive order in an arbitrary metric space. More precisely, given two random variables on a metric space, the order decides which one has a more concentrated distribution. It is motivated by the wish to be able to make new comparisons that the currently defined orderings cannot make. Related to this problem, several statistical parameters of a random variable, giving different kinds of information about the distribution, are studied. It is proved that the new dispersive order also compares some of these parameters. Special attention is paid to Riemannian manifolds, a natural class of metric spaces. Here, the new order behaves in a distinguished way: the differentiability axioms allow obtaining several equations which facilitate the decision. The principal properties of the order are presented, as well as several applications. Finally, the independence of this order from those already known is proved.

Forecast evaluations aim to choose an accurate forecast for making decisions by using loss functions. However, different loss functions often generate different ranking results for forecasts, which complicates the task of comparisons. In this paper, we develop statistical tests for comparing performances of forecasting expectiles and quantiles of a random variable under consistent loss functions. The test statistics are constructed with the extremal consistent loss functions of Ehm et.al. (2016). The null hypothesis of the tests is that a benchmark forecast at least performs equally well as a competing one under all extremal consistent loss functions. It can be shown that if such a null holds, the benchmark will also perform at least equally well as the competitor under all consistent loss functions. Thus under the null, when different consistent loss functions are used, the result that the competitor does not outperform the benchmark will not be altered. We establish asymptotic properties of the proposed test statistics and propose to use the re-centered bootstrap to construct their empirical distributions. Through simulations, we show the proposed test statistics perform reasonably well. We then apply the proposed method on (1) re-examining abilities of some often-used predictors on forecasting risk premium of the S&P500 index; (2) comparing performances of experts' forecasts on annual growth of U.S. real gross domestic product; (3) evaluating performances of estimated daily value at risk of the S&P500 index., 72 pages, 12 figures