The cumulative residual extropy has been proposed recently as an alternative measure of extropy to the cumulative distribution function of a random variable. In this paper, the concept of cumulative residual extropy has been extended to cumulative residual extropy inaccuracy (CREI) and dynamic cumulative residual extropy inaccuracy (DCREI). Some lower and upper bounds for these measures are provided. A characterization problem for the DCREI measure under the proportional hazard rate model is studied. Nonparametric estimators for CREI and DCREI measures based on kernel and empirical methods are suggested. Also, a simulation study is presented to evaluate the performance of the suggested measures. Simulation results show that the kernel-based estimator performs better than the empirical-based estimator. Finally, applications of the DCREI measure for model selection are provided using two real data sets.

We consider the moments and the distribution of hitting times on the lollipop graph which is the graph exhibiting the maximum expected hitting time among all the graphs having the same number of nodes. We obtain recurrence relations for the moments of all order and we use these relations to analyze the asymptotic behavior of the hitting time distribution when the number of nodes tends to infinity.

Decentralized consensus protocols have a variety of parameters to be set during their deployment for practical applications in blockchains. The analysis given in most research papers proves the security state of the blockchain, at the same time usually providing a range of acceptable values, thus allowing further tuning of the protocol parameters. In this paper, we investigate Ouroboros Praos, the proof-of-stake consensus protocol deployed in Cardano and other blockchains. In contrast to its predecessor, Praos allows multiple honest slot leaders that lead to fork creation and resolution, consequently decreasing the block rate per time unit. In our analysis of dependence on protocol parameters such as active slot coefficient and p2p network block propagation time, we obtain new theoretical results and explicit formulas for the expectation of the length of the longest chain created during the Praos epoch, the length of the longest unintentional fork created by honest slot leaders, the efficiency of block generation procedure (the ratio of blocks included in the final longest chain vs the total number of created blocks), and other characteristics of the blockchain throughput.

We study these parameters as stochastic characteristics of the block generation process. The model is described in terms of the two-parametric family ξij of independent Bernoulli random variables which generate deformation of the binomial distribution by a positive integer parameter—the delay (deterministic or random). An essential part of our paper is a study of this deformation in terms of denumerable Markov chains and generating functions.

Conditional risk measures and their associated risk contribution measures are commonly employed in finance and actuarial science for evaluating systemic risk and quantifying the effects of risk interactions. This paper introduces various types of contribution ratio measures based on the multivariate conditional value-at-risk (MCoVaR), multivariate conditional expected shortfall (MCoES), and multivariate marginal mean excess (MMME) studied in [34] (Ortega-Jiménez, P., Sordo, M., & Suárez-Llorens, A. (2021). Stochastic orders and multivariate measures of risk contagion. Insurance: Mathematics and Economics, vol. 96, 199–207) and [11] (Das, B., & Fasen-Hartmann, V. (2018). Risk contagion under regular variation and asymptotic tail independence. Journal of Multivariate Analysis 165(1), 194–215) to assess the relative effects of a single risk when other risks in a group are in distress. The properties of these contribution risk measures are examined, and sufficient conditions for comparing these measures between two sets of random vectors are established using univariate and multivariate stochastic orders and statistically dependent notions. Numerical examples are presented to validate these conditions. Finally, a real dataset from the cryptocurrency market is used to analyze the spillover effects through our proposed contribution measures.

The problem of reconstructing a distribution with bounded support from its moments is practically relevant in many fields, such as chemical engineering, electrical engineering, and image analysis. The problem is closely related to a classical moment problem, called the truncated Hausdorff moment problem (THMP). We call a method that finds or approximates a solution to the THMP a Hausdorff moment transform (HMT). In practice, selecting the right HMT for specific objectives remains a challenge. This study introduces a systematic and comprehensive method for comparing HMTs based on accuracy, computational complexity, and precision requirements. To enable fair comparisons, we present approaches for generating representative moment sequences. The study also enhances existing HMTs by reducing their computational complexity. Our findings show that the performances of the approximations differ significantly in their convergence, accuracy, and numerical complexity and that the decay order of the moment sequence strongly affects the accuracy requirement.

In this work, we focus on stochastic modeling for sustainable systems and introduce the family of r-modified reliability systems. This new family generalizes classical reliability systems studied in the literature by considering the components in the system to exhibit a kind of dependence that relaxes the component operating requirements and provides energy and resource efficiency. From a theoretical viewpoint, such a dependence is modeled with the use of a modified binary sequence. We then derive the reliability of two members of the family, i.e., the r-modified-k-out-of-n:F system and the r-modified-consecutive-k-out-of-n:F system, under different assumptions on the component reliabilities by using a variety of approaches, including Markov chains, combinatorial methods, and simple probabilistic arguments. We finally give some examples of real-life systems wherein the developed models and results are applicable and present the corresponding numerical results.