Existing outliers always make it difficult to estimate the dependence parameter of bivariate variables. Parameter estimation plays a major role in statistical inference and applied statistics. Bivariate copula is one of the effective tool to study the outliers because of its simplicity. In this article, we use bivariate copula and obtain a formula for it in presence of outliers, as a prevalent way. We consider $n$ sample pairs of $(X_i,Y_i)$, in which we have $k$ pairs of outliers and $(n-k)$ pairs of real data and introduce their likelihood function in presence of $k$ outliers and then estimate the corresponding dependence parameter $\theta$ and the noise parameter $\beta$, using some different methods via copula function. Also, some measures of dependence are obtained and compared in presence of outliers empirically. Finally, FGM copula and its application to real case study are considered for illustrating the results.

Keywords