Adding parameters to a known distribution is a valuable way of constructing flexible families of distributions. In this paper, we introduce a new Model, the Modified Additive Hazard Rate Model, by replacing the Additive Hazard Rate distribution in the general proportional add ratio Model. Next, when two sets of random variables follow the Modified Additive Hazard Model, we establish stochastic comparisons between the series and parallel systems comprising these components.

Background & Objectives: The analysis of age and survival probabilities, today, is several decades old. The calculation of age average in different groups of human society and analysis of factors which are effective on it to access methods for increase of individual’s health and age is one the daily concerns of scientists in different branches of biology and medicine. The objectives of this study were to assess the effects of prognostic factors on survival of patients with breast cancer using the Aalen Additive Hazards Model, and to illustRate the advantage of Aalen’s plot.Materials & Methods: during 2005-2007, 133 women with breast cancer were treated in Fayazbakhsh hospital, Tehran. All patients followed until 2011 April. To analysis data, Aalen’s Additive Hazard Model used. The graph of estimated cumulative regression function (Aalen s graph) was draw to analyze the effect of independent variable during time.Results: Significant factors in univariate analysis was entered into multivariable analysis of Allen`s Additive Hazard Model.HER2, third tumor malignancy grade and the number of positive lymph nodes more than10 were recognized as significant factors in patient`s survival in multivariable Model.Conclusion: Additive Hazard Model is used as a suitable substitution for Cox Model. In which standard deviation of estimations will be smaller than Cox Model.

Background and Objectives: Colorectal cancer is the fourth leading cause of cancer death worldwide. Annually 1430000 cases would be diagnosed and half of these people would die of the disease. In Iran, colorectal cancer is the fifth common cancer among men and is the thirdamong women. This study was designed to estimate survival Rates for patients with colorectal cancer and related factors.Methods: During the 1384 to 1388 years 24, 807 patients diagnosed with colorectal cancer were recorded in the Cancer Registry of the Ministry of Health and Medical Education. Age, gender, marital status, education, occupation, place of residence, tumor diagnosis, the clinical and demographic factors in colorectal cancer as effective on survival entered in Aalen’s Additive Hazard Model and diagram Aalen's was used to study the effects of these factors over time. R 3.2.0 software was used for data analysis.Results: Stage of tumor had a significant association with survival Rates. In this study 1 to 5 year survival Rate for patients with stage II tumors, was 100%, 98%, 96%, 93% and 93% respectively and for patients with stage III tumors 99%, 97%, 94%, 90% and 84%, respectively.Conclusion: In spite of using Cox Model in survival analysis by many researchers, Aalen’s Model may yield new insights in prognostic studies of survival time of patients with colorectal cancer over time. Our results suggest that early detection of patients in primary stage is important to increase survival.

Background & Objective: In most countries, heart diseases are known as first cause of death among non-contagious diseases and in Iran it is one of the most important causes of death. Therefore assessing risk factors associated with survival of these patients is necessary. In this study, effect of some risk factors on Hazard of death of subjects with acute myocardial infarction (AMI) is investigated.Materials & Methods: In this historical cohort study, 197 patients that experience AMI for the first time was studied which were admitted in Bushehr hospitals during 1997-2001. Aalen’s Additive Hazard Model used in this study to assess the relation of death Hazard Rate after first AMI and risk factors like age at time of infarction, patient’s sex, systolic and diastolic blood pressure, serum total cholesterol, LDL and HDL.Results: Average survival time of patients in the present study was more than 4 years. The result showed that there is no significant relation between sex and survival time of patient (p>0.05) however age at time of infarction has a significant relation with survival of patient (p<0.05).Conclusion: According to Aalen’s Additive Hazard Model as age at time of infarction increases, survival risk of AMI patients increases.

Introduction Fail-safe systems ((n 􀀀,1)-out-of-n Systems) are commonly used in many day-to-day applied structures. A fail-safe is a special design feature that will respond when a failure occurs so that no harm happens to the system itself. The brake system in a train is an excellent example of a fail-safe system in which the brakes are held in off-position by air pressure. If a brake line splits or a carriage becomes sepaRated, the air pressure will be lost,in that case, the brakes will be applied by a local air reservoir. However, another classic example of a fail-safe system is an elevator in which brakes are held off brake pads by tension, and if the tension gets lost, the brakes latch on the rails in the shaft, thus preventing the elevator from falling off. There are many other such fail-safe systems in common use. Balakrishnan et al. (2015) established necessary and sufficient conditions for comparing two fail-safe systems with independent homogeneous exponential components in terms of mean residual life, dispersive, Hazard Rate and likelihood ratio orders. Their results specifically showed how an (n−, ١, )-out-of-n system consisting of heterogeneous components with exponential lifetimes could be compared with any (m−, ١, )-out-of-m system consisting of homogeneous components with exponential lifetimes. Similarly, Zhang et al. (2019) presented sufficient (and necessary) conditions on the lifetimes of components and their survival probabilities from random shocks for comparing the lifetimes of two fail-safe systems in terms of standard stochastic, Hazard Rate and likelihood ratio orders. Cai et al. (2017) compared the Hazard Rate order of second-order statistics arising from two sets of independent multiple-outlier proportional Hazard Rates (PHR) samples. Material and Methods The comparison of essential characteristics associated with lifetimes of technical systems is an exciting topic in reliability theory since it usually enables us to approximate complex systems with simpler systems and subsequently obtain various bounds for important agreeing characteristics of the complex system. A convenient tool for this purpose is the theory of stochastic orderings. Results and Discussion This paper discusses the Hazard Rate order of (n􀀀, 1)-out-of-n systems arising from two sets of independent multiple-outlier Modified proportional Hazard Rates components. Under certain conditions on the parameters and the submajorization order between the sample size vectors, the Hazard Rate order between the (n􀀀, 1)-out-of-n systems from multiple-outlier Modified proportional Hazard Rates is established. Conclusion In this paper, we have presented sufficient conditions for the Hazard Rate order between fail-safe systems. It will be of great interest to generalize the current work from lifetimes of fail-safe systems to those of k-out-of-n. Another problem of interest will be to consider the setting of general systems with several subsystems having dependent components and extend the results established here to this general case. We are currently working on these problems and hope to report those findings in a future paper.

In some applied problems we need to choose a population from the given populations and estimate the parameter of the selected population. Suppose k random samples are chosen from k populations with proportional Hazard Rate Model or proportional reversed Hazard Rate Model. According to a specified selection rule, it is desired to estimate the parameter of the best (worst) selected population. In this paper, under the entropy loss function we obtain the uniformly minimum risk unbiased (UMRU) estimator of the parameters of the selected population, and derived sufficient conditions for minimaxity of a given estimator.Then we find the class of admissible and inadmissible linear estimators of the parameters of the selected population and determine the class of dominators of a given estimator. We show that the UMRU estimator is inadmissible and compare the obtained estimators by plotting their risk functions.

In this paper, we discuss the prediction problem based on censored coherent system lifetime data when the system structure is known and the component lifetime follows the proportional reversed Hazard Model. Different point and interval predictors based on classical and Bayesian approaches are derived. A numerical example is presented to illustRate the prediction methods used in this paper. Monte Carlo simulation study is performed to evaluate and compare the performances of different prediction methods.

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 paper, we provide sufficient and necessary conditions on components' lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two $(n-1)$-out-of-$n$ systems in two cases: (i) when components are independent, and then (ii) when components are dependent.