Abstract
In multivariate survival analysis, estimating the multivariate distribution functions and then measuring the association between survival times are of great interest. Copula functions, such as Archimedean Copulas, are commonly used to estimate the unknown bivariate distributions based on known marginal functions. In this paper the feasibility of using the idea of local dependence to identify the most efficient copula model, which is used to construct a bivariate Weibull distribution for bivariate Survival times, among some Archimedean copulas is explored. Furthermore, to evaluate the efficiency of the proposed procedure, a simulation study is implemented. It is shown that this approach is useful for practical situations and applicable for real datasets. Moreover, when the proposed procedure implemented on Diabetic Retinopathy Study (DRS) data, it is found that treated eyes have greater chance for non-blindness compared to untreated eyes.
Highlights
A copula function is a rule which gathers or couples one-dimensional marginal distribution functions into a form of multivariate distribution function
Fisher [1], discussed the importance of copula precisely in his transcripts in the Encyclopedia of Statistical Sciences, “Copulas are of interest to statisticians for two main reasons; first, as a way of studying scale-free measures of dependence; and secondly, as a starting point for constructing families of bivariate distributions”
Three different models of Archimedean have been considered to derive bivariate Weibull distribution (BWD); Gumbel copula, Clayton copula and Independent copula, with association parameter θ which is given by Kendal tau (τ) [2]
Summary
A copula function is a rule which gathers or couples one-dimensional marginal distribution functions into a form of multivariate distribution function. In order to identify the best copula, a correlate Weibull random variable is generated to compute the Local Dependence for these copulas. Three different models of Archimedean have been considered to derive bivariate Weibull distribution (BWD); Gumbel copula, Clayton copula (aka Cook and Johnson’s copula) and Independent (or Product) copula, with association parameter θ which is given by Kendal tau (τ) [2].
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