The Estimation And Application Of Copula Function

The research on the dependence of multiple variables, has a very importantsignificance in the field of financial risk analysis and avoid operational risk loss. Basedon this, Copula theory has been introduced to the analysis of the correlation betweeneconomic variables. Using copula theory analysis the related structures betweenvariables, one of the key issues is the estimation of Copula density function. In general,according to the statistical characteristics of sample data, choose suitable copulas model,and then to estimate the unknown parameters. Maximum likelihood estimation method,the step-by-step estimation method and the semiparametric estimation method are thecommonly used parameter estimation methods of the Copula density function.Based on Copula theory, the bayesian parameter estimation method and thenonparametric kernel density estimation method of the copulas density function has beenintroduced. And extend the bayesian parameter estimation method to the multi-dimensional case, both the theory of narration, and the analysis of the actual data. For thebayesian parameter estimation of multivariate normal Copula, the key is to estimate thecorrelation coefficient matrix, in which the selection of the prior distribution is adifficulty. To solve this problem, the approach taken in this article is to estimate thecovariance matrix. Select the Wishart distribution as the conjugate prior distribution forthe inverse of the covariance matrix. Based on the posterior distribution and Gibbssampling, taking bayesian estimation under square loss function which is the averages ofthe posterior distribution, as the estimates of the covariance matrix. Then according tothe relationship of the correlation coefficient matrix and covariance matrix, calculates theparameter estimation value of the Copula model.In addition to the bayesian parameter estimation method, the paper also proposednonparametric kernel density method to estimate the copula density function. This method does not make any prior assumptions on the copula function, more flexibleapplication. The key is the choice of window width when discussing kernel densityestimation method for Copula. A plug-in method for the choice of the window width isput forward in this article,which does not involve any reference code,only use sampledata to solve the problem completely.The simulation and empirical analysis has been carried on for validation of thefeasibility and accuracy of the bayesian parameter estimation method and non-parametrickernel density estimation method. Results show that the method using bayesianparameter estimate Copula function, the sampling trajectory of the parameters is stableand given to illustrate the convergence of the sequence, then the estimated value of theparameter can be thought of as very accurate; From the figure of the Copula densityfunction with the nonparametric kernel density estimation, it can be seen kernel densityestimation method able to characterize the structure of the correlation between variables,which starting directly from the sample data to analyze the correlation structure,accurately fitting the characteristics of the data itself, relatively close to the actualsituation.