Ordinary differential equations(ODEs) with stochastic processes in their vector field, have lots of applications in science and engineering. The main purpose of this article is to investigate the numerical methods for ODEs with Wiener and Compound Poisson processes in more than one dimension. Ordinary differential equations with Ito diffusion which is a solution of an Ito stochastic differential equation will be considered. Because for the numerical solution of these equations we need the simulation of stochastic double integrals, we explain the simulation of these integrals in more details. Also one-step and multi steps methods for the solution of affine random ordinary equations (RODEs) which are an important class of RODEs will be considered. The numerical solution of these equations with Wiener and Compound Poisson processes will be established. Two methods for simulation of the double integrals will be explained, and some numerical examples are provided to confirm the theoretical results numerically.

Background & Aim: The excess hazard rate proposed by Andersen and Vaeth may underestimate the long-term excess hazard rate for cancer survival. Zahl explained the phenomenon by continuous selection of the most robust individuals after diagnosis. He applied correlated inverse Gaussian and gamma frailty models to estimate excess intensity and reached a better estimate of the rate and called it the corrected excess hazard. The compound Poisson distribution has more parameters and therefore owns more flexibility and includes gamma and inverse Gaussian distributions as special cases. Therefore, the aim of this study was to estimate the excess hazard using compound poisson frailty model Methods & Materials: Both shared and correlated frailty (CF) variables based on compound Poisson distribution were used to model unobserved common covariates. A data set of patients diagnosed with localized or regional gastrointestinal tract cancer collected at the Mazandaran province of Iran was studied. As registration systems in Iran are so affected by omission and various errors, a number of five West Coale- Demeny life tables for men and four for women were constructed corresponding to each birth cohort, which was considered as the reference life tables. Thus, population-based mortality rates [h1 (t)] were simply replaced by the appropriate values of the West tables depending on the sex (male or female) and birth cohort of the patient.Results: The CF model with unequal variances could best estimate the long-term excess hazard.Conclusion: This study advocates the CF models can best estimate the long-term excess hazard rates regardless of the distribution of the frailty variable.

Nonhomogeneous Poisson processes (NHPPs) are often used to model recurrent events, and there is thus a need to check model fit for such models. We study the problem of obtaining exact goodness-of-fit tests for certain parametric NHPPs, using a method based on Monte Carlo simulation conditional on sufficient statistics. A closely related way of obtaining exact confidence intervals in parametric models is also briefly considered.

This paper focuses on different methods of estimation and forecasting in first-order integer-valued autoregressive processes with Poisson-Lindley (PLINAR(1)) marginal distribution. For this purpose, the parameters of the model are estimated using Whittle, maximum empirical likelihood and sieve bootstrap methods. Moreover, Bayesian and sieve bootstrap forecasting methods are proposed and predicted value for h-step ahead of the series is obtained. Some simulations and a real data analysis are applied to compare the presented estimations and the prediction methods.

This research at making a comparison between phonological processes in complex and compound Persian words. Data are gathered from a 40, 000-word Persian dictionary. To catch some results, 4, 034 complex words and 1, 464 compound ones are chosen. To count the data, "excel" software is used. Some results of the research are: 1- "Insertion" is the usual phonological process in complex words...

The non-homogeneous bivariate compound Poisson process with short term periodic intensity function is used for modeling the events with seasonal patterns or periodic trends. In this paper, this process is carefully introduced. In order to characterize the dependence structure between jumps, the Lé vy copula function is provided. For estimating the parameters of the model, the inference for margins method is used. As an application, this model is fitted to an automobile insurance dataset with inference for margins method and its accuracy is compared with the full maximum likelihood method. By using the goodness of fit test, it is confirmed that this model is appropriate for describing the data.

In this article, three monitoring approaches using cumulative sum (CUSUM) control charts in phase two for zero inflated poisson-based processes are presented. The first approach is based on the zero inflated poisson distribution, the second on, a proportional hazard regression model, and the third integrates a proportional hazard (PH) regression model and a frailty model to consider both measurable and unmeasurable covariates. The performance of all three control charts was evaluated separately and simultaneously by applying shifts to both parameters of the zero inflated poisson distribution. Extensive simulation studies were conducted to evaluate the performance of these monitoring methods in terms of the average run length (ARL) of the control charts. The proposed cumulative sum control chart with simultaneous consideration of measurable and unmeasurable variables showed superior performance. Finally, a real case study in a label printing factory has been provided to show the effectiveness of the proposed control chart.

Nowadays, most of products are the results of different dependent process steps. Due to the cascade property in most of these processes, using the traditional control charts for monitoring these processes is not suitable. To solve this problem, Cause selecting Charts (CSCs) are developed to monitor multistage processes. These control charts have usually been developed when quality characteristics follow normal distribution. However, sometimes other distributions rather than normal can characterize quality characteristics. In this paper a cause selecting control chart based on the standardized residuals of a generalized linear model is developed to monitor a two-stage process with a Poisson distributed quality characteristic in the second stage. The performance of the proposed control chart is investigated in terms of average run length criterion under two different link functions in comparison with the best method in the literature. The results show the better performance of the proposed control chart in detecting increasing shifts. Finally, the performance of the proposed method in practice is evaluated through a case and acceptable results are obtained.