Copulas are functions that link multivariate distribution function to marginal distribution functions and separate marginal distribution from dependency structure. Copulas can be used as suitable tool for modeling dependent variables. But there are some restrictions in modeling data and construction methods using copula functions.
In some copula functions, because of limited dependency domain, to model high dependency variables is not possible. For example, about FGM copula, one of the most applicable copulas, dependency range is limited. Also, the copula functions consider certain dependency structure while we may face with a weight of dependency structures and more important than two former cases, symmetry of copula function; Symmetry of copula functions state that the interested variables play the same role in their joint distribution function, while this precondition, apart from restricting the application domain of the copula function, cannot describe some models.
In this thesis, to remove mentioned limitations, an extension of the FGM copula in terms of polynomial section with degrees of to improve dependency domain based on extreme value copula is introduced. Also, to predominate symmetry restriction in copula functions, an asymmetric class from introduced generalization will be presented. In sequel, another asymmetric class of generalized GB copula in terms of the definition of the different function from uniform marginal distribution function is introduced, whose some of properties will be presented and generated sub-families are evaluated by modeling the risk of diabetes as application and simulation. finally, in order to upgrade the certain dependency structure, a generalization of Clayton copula under weighted bivariate functions with weighted dependency structure will be presented and its applications in hydrology science will be surveyed. Also, we study measurements and dependency concepts in introduced classes. Generally, generalization of copula functions is an approach in introducing classes of copula functions that leads to extract new properties of copula function. These generalizations can be applied to generate new sub-families of copulas and improve domain of dependency measures. Also, generated sub-families from these generalizations are applicable in modeling dependent data in different fields of science. Generating copulas is useful in constructing bivariate distributions with different marginal distribution functions and weighted bivariate distributions.

 

Examining the dependency structure and constructing multivariate distributions are a vital subject in analyzing uncertainty between varies phenomena. Copula function link marginal distribution and try to discover the dependency structure and build multivariate distribution. Finding suitable copula function is always a concern such that this challenge highlighted by increasing the dimension of variables and restriction in constructing multivariate copula. In other words, difficulty in determining multivariate distribution translates one difficult problem into another. Lack of suitable goodness of fit test in selecting optimal copula faced this problem with more difficulty. To settle this concern, nonparametric copula introduced. Empirical copula introduced as one of first nonparametric copulas. It is highly discontinuous, not of much use in practice. Kernel-based approach proposed as another nonparametric copula that enjoys good properties but their computations entail non-trivial implementation issues. Also, bandwidth selection more discussed in the literature such that there is not agreement about it. It is proposed another nonparametric method to estimate copula density functions based on two dimensional orthonormal series expansion which depends on selecting cut point. The main downfall of these bases as spline method is their infinite support which demanding a large number of terms in the series expansion. The advent of wavelets enhanced using orthonormal series for copula density estimation. In statistical methodologies, density estimation and copula density estimation using wavelet introduced as suitable nonparametric estimation. A known characteristic of wavelet functions is that they cannot be symmetric, orthogonal and compact support at the same time while multiwavelets overcome this disadvantage. These properties highlight the usefulness of the multiwavelet in order to approximate copula density functions. \\ In this thesis, we estimate copula density using multiwavelets as vector-valued type of wavelets and use as nice nonparametric estimation of copula density. Multiwavelets by Possessing three appropriate properties at the same time, high smoothness and high approximation order properties, can be more precise in copula density approximation. Due to orthogonality properties in multiwavelet, approximation coefficients could be easily determined. Compact support property in multiwavelet causes a good level of approximation achieves with less terms and symmetric property leads to a nice approximation in local and global symmetry. The presented approximation could be followed using different multiwavelets. We evaluate our approximation method using the mean integrated square error (MISE), the point wise error (PWE) and the AIC criteria. Finding results reveal that copula density approximation is better than scalar wavelet. We make this approximation method more accurate by using multiresolution analysis that balance between using terms and desired approximation. According to the finding results, in low resolution level, multiwavelet estimate copula density better that wavelet and converge to the underlying copula faster than wavelet. We simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data. Moreover, we apply Legendre multiwavelet as a polynomial type of multiwavelets with simple closed form in order to approximate copula density functions such that its powerfull smoothness provides a nice approximation. The support of Legendre multiwavelet is defined on interval $[0,1]$ as copulas that are normalized to have the support on the unit square and uniform marginal. This is a reason that the integration of the estimated copula density has been always one and do not require correction. We then apply presented method, estimating copula density through multiwavelets, to approximate the multivariate copula density and multivariate distribution using pair-copula as a flexible graphical model. Finally, we apply presented method to model a dataset of financial and actuarial data.

In reliability studies and lifetime testing experiments, we may encounter instances in which the lifetime of units has been reported as imprecise quantities. The classical statistical methods for estimating the parameters of lifetime distributions are not appropriate to deal with such imprecise cases. In such circumstances, using the theory of fuzzy sets, such imprecise data can be modeled with fuzzy sets. In this thesis, the parameter estimation of lifetime distributions based on fuzzy data is studied. Exponential distribution, Rayleigh distribution and Weibull distribution are survival models which are popular due to their flexibility and have many applications in reliability analysis. In the research literature, estimating the parameters of these modeles based on maximum likelihood and Bayesian methods have been studied. In this thesis, generalization of these methods for estimating the lifetime distribution parameters based on fuzzy data will be done.
In many lifetime testing studies, if the experimenter wants to observe and record the failure times of all units, he must consider the time of experiment as the maximum lifetime of units in the test, that in most cases it is virtually impossible. Therfore, we deal with samples in which complete information of sample units are not recorded for some reasons.We define this constraints arising in the sample as censor. The most common types of censoring schemes in the studies are: type I censoring, type II censoring, progressive type I censoring, progressive type II censoring, hybrid type I censoring, hybrid type II censoring and hybrid progressive type II censoring. Statistical inference based on these censoring plans have been considered by many statisticians. The research results presented in the studies are based on the assumption that the observed lifetime data are crisp values, but in most practical applications we may be faced with imprecise data. In this thesis, we consider the problem of estimating the parameter of lifetime distributions based on imprecise censored data. We first model the imprecise observations of life testing experiments with fuzzy sets and present a new generalization of the likelihood function under different censoring schemes. Assuming that the lifetime distributions of the test units are exponential and Rayleigh distributions, we obtain the estimation of the distribution parameters based on maximum likelihood and Bayesian approaches. Then, we examine the performance of proposed methods using simulation studies and practical examples.
In the discussion of statistics, comparisons between quantitative and different societies have always been an important place. Essentially, a substantial part of statistical issues is devoted to introducing methods of comparison. This subject has a special importance in survival analysis and reliability theory and a variety of comparison methods have been studied by researchers. Using probabilistic models of stress-strength, , is one type of comparison between the two variables that has received much attention in recent decades. According to the definition of the reliability of a system, one can say that referes to the reliability of a system in which random variables and are the stress and strength of the system. Parameter estimation of based on maximum likelihood and Bayesian approaches, when and are statistically independent random variables and their observations are reported in the form of imprecise quantities, are the other objectives of this thesis.