Survival analysis is one of the extensive and high-usage branches of statistics. The purpose of survival analysis is usually computing the ‎hazard‎ of interested event and, thus, survival probability at any time. Survival analysis is a set of models and methods that each of them can be used for data analyzing according to the exiting status. Multivariate survival and recursive event data are different types of data used in survival analysis, and it is assumed that by the end of the study, all subjects under study experiences the intended event. But sometimes an interesting event for everyone may not occur, as a result, these people are considered as cured, protected and/or insusceptible. Models that account for part of a cured society are usually named Long-term or Cured models. most popular cured models is mixed cure model. To account for association between survival times, we use copula function. The primary objective of this thesis is to achive some flexible estimate of baseline hazard based on the cox proportional hazard model using piecewise constant semi-parametric approach. The secondary objective is the modeling of multivariate survival and recursive event data using Copula function with piecewise exponential marginal distributions. These models are respectively applied to corneal transplantation data and birth interval data in Ahwaz city. In the following, we study the cured models and once again we analyze the data sets relevant to corneal transplantation using these models. Deviance Information Criterion is used to compare the resulting models.

 survival analysis is one of the most papular branches in statistics. in survival analysis or, more generally in time to event analysis the aim is calculation the risk of reccurence event or survival probability at any time. COX proportional hazard medel is one of the most widely used of survival models where by this model we can mesure the effects of covariates on time to event. but one of parts PH model is basline hazard named. in this dissertation our objects obtained the flexible and smooth estimation for basline hazard in PH model using beysian P-splines. Also in frialty models where one of the cox PH model exponsions where used in situations that survival times are not independent too estimation the baseline hazard function using beysian P-splines and two real data sets analysed by this methods

 In the subject of Stress – Strength models, our aim is evaluation the probability of (Reliability parameter) where Y is Stress component and X is Strength component. For that the component keep doing this job, the Stress must not exceed the Strength. In the past many subjects have been discussed about Reliability parameter when X and Y are independent, but, unfortunately when these two component are correlated have been less discussed. In this thesis, our goal estimation Reliability parameter under the assumption of dependence between the two Stress and Strength component. In this thesis, will estimation the Reliability parameter for Standard Weibull distribution, Generalization Exponential distribution and Kumaraswamy distribution based methods through copula function which was by the Domma and Giordano introduced. We give the close form for Reliability parameter under Farlie-Gumbel-Morgenstern copula and generalization, but because of nonexistence of close form for Reliability parameter under Frank copula, we will use numerical routines to obtain it. We will use the simulation in order to demonstrate suitability of the method used to estimate the Reliability parameter. In application we used the reliability parameter for internal and external real data.

The main goal of many clinical studies is investigate the relationship between survival time and risk factors. Being independent and identically distribution of observations survival time mean Which are homogeneous population, but in some studies, especially studies done on humans may be unknown factors other than the covariates, that distribution of survival time and, subsequently, strongly affect the hazard function. But because of the unknown or the inability to measure them are not able to include them in hazard regression.
In the survival cluster data, the cluster effects of unobserved may affect the results, leading to dependence among individuals in the same cluster.
To resolve this problems and take into account between Individuals and observed events, to be used generalization of Cox model is fraility. In this thesis, in addition to investigate the factors affecting on the birth interval, also use the random effect (effect fraility) for analysis from birth to live. Here use survival multilevel model (model fraility) with two randomly (health center) and the person (mother). The results show that the two randomized effects into the model, the random effect of the mother on the birth is significant. This random effect led to the correlation between birth interval.

Multivariate survival data are used in many different scientific fields. To consider the dependence structure among survival times, there are different methods. One approach that has received considerable attention in recent years is using copula function for modeling frailty function. Copulas are functions that connects the marginal distributions to restore the joint distribution. Survival analysis assumes that all the subject experience event under Study, but may the event of interest does not occur for all the subject. These individuals are known as cured, non – susceptible and immune. Models that consider part of the population cured usually are called long – term survival models or cured models. The most common type is a mixture cure model. In this thesis, the aim is modeling the long – term bivariate using copula. Therefore copula functions Archimedes Clayton, Gumbel, Frank and copula function Fariley - Gumbel - Morgenstern were used.
Sometimes observation of survival time was obtained from independent cluster. The reason of this independent is unmeasured covrate in unseen determined cluster.
So independent was intered with random effect in mixture cure model. In the end of we modeled mixture cure models combined copula function on bilateral corenal graft data and cure models combined copula function with random effect on Ahwaz birth interval obtained clusters.To choose the best fitted model to the data was used standard deviation information (DIC).