In this paper, the complete f-moment convergence for weighted sums of sequences of extended negatively dependent random variables is investigated. As a result, we not only extend and generalize some results on complete convergence obtained by Wu and Zhai (Journal of Mathematical Inequalities, 13(1): 251-260, 2019), but also get some results on the complete consistency for the estimator of nonparametric regression models and provide some simulations to assess the validity of our theoretical results.

In this study, almost sure asymptotic properties of the maximum likelihood estimator in an autoregressive process driven by stationary Gaussian noise are obtained. Precisely, we show that the maximum likelihood estimator is strongly consistent whereas it is well known that the least-square estimator may lack consistency, especially when the process is driven by a correlated noise. Furthermore, the local asymptotic normality of the likelihood ratio is established in order to build an asymptotically uniformly invariant most powerful procedure for testing the significance of the autoregressive parameter. Finally, we illustrate our results in a short simulation study.

In this paper, we introduce a bivariate distribution on arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.

In this paper, we derive stochastic Leibniz formulas for Volterra integrals under enlarged filtrations. We investigate both pure-jump and Brownian Volterra processes under diverse initially enlarged filtration approaches. In these setups, we compare the ordinary with the stochastic (Doléans-Dade) exponential of a Volterra process and provide the corresponding martingale conditions. We also consider backward stochastic Volterra integral equations (BSVIEs) under enlarged filtrations and obtain the related solution formulas. We finally propose an anticipative Heath Jarrow Morton (HJM) forward rate model of Volterra-type and infer the associated bond price representation. In an introductory section, we compile various facts on deterministic and stochastic Leibniz formulas for parameter integrals.

In reliability literature, shock models and multi-state systems have both been discussed extensively. Shock models have attracted a great deal of attention due to their important role in engineering systems. In the δ-shock model, a system fails if the interval time between two successive shocks is less than a pre-fixed threshold δ. In this study, a new version of mixed δ-shock models is defined by combining δ-shock and extreme shock models, thus causing the failure of a multi-state system in two ways: first, when k interarrival times between two consecutive shocks with magnitude larger than the critical threshold γ are in and second, when the interarrival time between two consecutive shocks is less than δ 1. The survival function is obtained for three cases: (i) system’s lifetime, (ii) when the system performs completely, and (iii) when the system performs partially. A Monte Carlo simulation study is carried out to validate the analytical results established here.

In this article, an extended mixed shock model is investigated, in which the system suffers from mutually dependent competing soft failure and hard failure processes. The system studied degrades gradually with time and the number of shocks coming from the environment. A soft failure occurs when the overall degradation amount surpasses the soft failure threshold of the system. The extended mixed shock model is a combination of a system-state-dependent extreme shock model and a system-state-dependent δ shock model. A hard failure occurs when the magnitude of a single shock exceeds the threshold D(t), or an inter-arrival time between shocks is less than the recovery time threshold The mutual dependency between the soft failure and hard failure processes is reflected in the fact that: (1) the external shock will accelerate the degradation process of the system by causing an abrupt increment of its degradation amount; (2) the system degradation amount determines the thresholds of the extreme shock model and the δ shock model. The system fails when a hard failure or a soft failure occurs. By using the stochastic process theory, the reliability expression and some reliability indices of the system are obtained. Some special cases of the model are also discussed. Finally, a case study of the pier columns of a sea bridge is presented to illustrate the developed model.

We consider a continuous-time branching random walk on a multidimensional lattice in a random branching medium. It is theoretically known that, in such branching random walks, large rare fluctuations of the medium may lead to anomalous properties of a particle field, e.g., such as intermittency. However, the time intervals on which this intermittency phenomenon can be observed are very difficult to estimate in practice. In this paper, branching media containing only a finite and non-finite number of branching sources are considered. The evolution of the mean number of particles with a random point perturbation and one initial ancestor particle at a lattice point is described by an appropriate Cauchy problem for the evolutionary operator. We review some previous results about the long-time behavior of the medium-averaged moments for the particle population at every lattice point as well as the total one over the lattice and present an algorithm for the simulation of branching random walks under various assumptions about the medium, including the medium randomness. The effects arising in random non-homogeneous and homogeneous media are then compared and illustrated by simulations based on the potential with Weibull-type upper tail. A wide range of models under different assumptions on a branching medium, a configuration of branching sources, and a lattice dimension were considered during the comparison. The simulation results indicate that intermittency can be observed in random media even over finite time intervals.

In this paper, we define a discrete time version of the nonhomogeneous Poisson process and study its properties. For the definition and analysis, we suggest new general concepts for discrete time point processes. The proposed process possesses several stochastic properties which are analogous to those of continuous time nonhomogeneous Poisson process. It will also be shown that, under certain conditions, the proposed discrete time version of nonhomogeneous Poisson process converges to the continuous time version. Based on the proposed process, useful reliability applications are discussed.

This note investigates where is a standard Brownian motion, with We derive explicit formula for the probability and find its asymptotic behavior both in the so called many-source and high-threshold regimes.

In this article, we consider residual lives of series-parallel and parallel-series systems with independent subsystems consisting of dependent homogeneous components with the joint distribution of their lifetimes being modeled by an Archimedean copula. By considering two such systems with different numbers of components within each subsystem, we establish various stochastic orderings and aging results for the residual lives of series-parallel and parallel-series systems. The results established here are quite general. Several examples are presented to illustrate all the established results and their reliability implications.