3.5 Comparison of TDC Estimators

In this section we investigate the finite-sample properties of the introduced TDC estimators. One thousand independent copies of $ m=500,1000,$ and $ 2000$ i.i.d. random vectors ($ m$ denotes the sample length) of a bivariate standard $ t$-distribution with $ \theta=1.5,2,$ and $ 3$ degrees of freedom are generated and the upper TDCs are estimated. Note that the parameter $ \theta$ equals the regular variation index $ \alpha $ which we discussed in the context of elliptically-contoured distributions. The empirical bias and root-mean-squared error (RMSE) for all three introduced TDC estimation methods are derived and presented in Tables 3.6, 3.7, and 3.8, respectively.


Table: Bias and RMSE for the nonparametric upper TDC estimator $ \hat{\lambda}^{(1)}_U$ (multiplied by $ 10^3$). The sample length is denoted by $ m.$
Original $ \theta=1.5$ $ \theta =2$ $ \theta=3$    
parameters $ \lambda_U= 0.2296 $ $ \lambda_U= 0.1817$ $ \lambda_U=0.1161 $    
Estimator $ \hat{\lambda}^{(1)}_U$ $ \hat{\lambda}^{(1)}_U$ $ \hat{\lambda}^{(1)}_U$    
  $ \;\;$Bias (RMSE) $ \;\;$ $ \;\;$ Bias (RMSE) $ \;\;$ $ \;\;$ Bias (RMSE) $ \;\;$    
           
$ m=500$ $ 25.5$ ($ 60.7$) $ 43.4$ ($ 72.8$) $ 71.8$ ($ 92.6$)    
$ m=1000$ $ 15.1$ ($ 47.2$) $ 28.7$ ($ 55.3$) $ 51.8$ ($ 68.3$)    
$ m=2000$ $ \,\,\,8.2$ ($ 38.6$) $ 19.1$ ($ 41.1$) $ 36.9$ ($ 52.0$)    


Table: Bias and RMSE for the nonparametric upper TDC estimator $ \hat{\lambda}^{(2)}_U$ (multiplied by $ 10^3$). The sample length is denoted by $ m.$
Original $ \theta=1.5$ $ \theta =2$ $ \theta=3$    
parameters $ \lambda_U= 0.2296 $ $ \lambda_U= 0.1817$ $ \lambda_U=0.1161 $    
Estimator $ \hat{\lambda}^{(2)}_U$ $ \hat{\lambda}^{(2)}_U$ $ \hat{\lambda}^{(2)}_U$    
  $ \;\;$Bias (RMSE) $ \;\;$ $ \;\;$ Bias (RMSE)$ \;\;$ $ \;\;$ Bias (RMSE)$ \;\;$    
           
$ m=500$ $ 53.9$ ($ 75.1$) $ 70.3 $ ($ 88.1$) $ 103.1$ ($ 116.4$)    
$ m=1000$ $ 33.3$ ($ 54.9$) $ 49.1 $ ($ 66.1$) $ 74.8$ ($ 86.3$)    
$ m=2000$ $ 22.4$ ($ 41.6$) $ 32.9$ ($ 47.7$) $ 56.9$ ($ 66.0$)    



Table: Bias and RMSE for the parametric upper TDC estimator $ \hat{\lambda}^{(3)}_U$ (multiplied by $ 10^3$). The sample length is denoted by $ m.$
Original $ \theta=1.5$ $ \theta =2$ $ \theta=3$    
parameters $ \lambda_U= 0.2296 $ $ \lambda_U= 0.1817$ $ \lambda_U=0.1161 $    
Estimator $ \hat{\lambda}^{(3)}_U$ $ \hat{\lambda}^{(3)}_U$ $ \hat{\lambda}^{(3)}_U$    
  $ \;\;$ Bias (RMSE) $ \;\;$ $ \;\;$ Bias (RMSE)$ \;\;$ $ \;\;$ Bias (RMSE) $ \;\;$    
           
$ m=500$ $ 1.6$ ($ 30.5$) $ 3.5$ ($ 30.8$) $ 16.2$ ($ 33.9$)    
$ m=1000$ $ 2.4$ ($ 22.4$) $ 5.8$ ($ 23.9$) $ 15.4$ ($ 27.6$)    
$ m=2000$ $ 2.4$ ($ 15.5$) $ 5.4$ ($ 17.0$) $ 12.4$ ($ 21.4$)    


Regarding the parametric approach we apply the procedure introduced in Section 3.4 and estimate $ \rho$ by a trimmed empirical correlation coefficient with trimming proportion $ 0.05\%$ and $ \alpha \;(=\theta)$ by a Hill estimator. For the latter we choose the optimal threshold value $ k$ according to Drees and Kaufmann (1998). The empirical bias and RMSE corresponding to the estimation of $ \rho$ and $ \alpha $ are provided in Tables 3.9 and 3.10. Observe that Pearson's correlation coefficient $ \rho$ does not exist for $ \theta< 2.$ In this case, $ \rho$ denotes some dependence parameter and a more robust estimation procedure should be used (Frahm et al.; 2002).




Table 3.9: Bias and RMSE for the estimator of the regular variation index $ \alpha $ (multiplied by $ 10^3$). The sample length is denoted by $ m.$
Original $ \theta=1.5$ $ \theta =2$ $ \theta=3$    
parameters $ \alpha =1.5$ $ \alpha = 2$ $ \alpha =3$    
Estimator $ \hat{\alpha}$ $ \hat{\alpha}$ $ \hat{\alpha}$    
  $ \;\;$Bias (RMSE) $ \;\;$ $ \;\;$Bias (RMSE) $ \;\;$ $ \;\;$ Bias (RMSE) $ \;\;$    
           
$ m=500$ $ \,\,\,\,\,\,~2.2$ ($ 211.9$) $ -19.8$ ($ 322.8$) $ -221.9$ ($ 543.7$)    
$ m=1000$ $ -14.7$ ($ 153.4$) $ -48.5$ ($ 235.6$) $ -242.2$ ($ 447.7$)    
$ m=2000$ $ -15.7$ ($ 101.1$) $ -60.6$ ($ 173.0$) $ -217.5$ ($ 359.4$)    



Table: Bias and RMSE for the ``correlation'' coefficient estimator $ \hat{\rho}$ (multiplied by $ 10^3$). The sample length is denoted by $ m.$
Original $ \theta=1.5$ $ \theta =2$ $ \theta=3$    
parameters $ \rho =0$ $ \rho =0$ $ \rho =0$    
Estimator $ \hat{\rho}$ $ \hat{\rho}$ $ \hat{\rho}$    
  $ \;\;$ Bias (RMSE) $ \;\;$ $ \;\;$ Bias (RMSE) $ \;\;$ $ \;\;$ Bias (RMSE)$ \;\;$    
           
$ m=500$ $ \;\;\,0.02$ ($ 61.6$) $ -2.6$ ($ 58.2$) $ \;\;\,2.1$ ($ 56.5$)    
$ m=1000$ $ -0.32$ ($ 44.9$) $ \;\;\,1.0$ ($ 42.1$) $ \;\;\,0.6$ ($ 39.5$)    
$ m=2000$ $ \;\;\,0.72$ ($ 32.1$) $ -1.2$ ($ 29.3$) $ -1.8$ ($ 27.2$)    


Finally, Figures 3.4 and 3.5 illustrate the (non-)parametric estimation results of the upper TDC estimator $ \hat{\lambda}^{(i)}_U,\;i=1,2,3.$ Presented are $ 3\times 1000$ TDC estimations with sample lengths $ m=500,1000$ and $ 2000.$ The plots visualize the decreasing empirical bias and variance for increasing sample length.

Figure: Nonparametric upper TDC estimates $ \hat{\lambda}^{(1)}_U$ (left panel) and $ \hat{\lambda}^{(2)}_U$ (right panel) for $ 3\times 1000$ i.i.d. samples of size $ m=500,1000,2000$ from a bivariate $ t$-distribution with parameters $ \theta =2$, $ \rho =0,$ and $ \lambda ^{(1)}_U=\lambda ^{(2)}_U=0.1817$.

Figure: Nonparametric upper TDC estimates $ \hat{\lambda}^{(3)}_U$ for $ 3\times 1000$ i.i.d. samples of size $ m=500,1000,2000$ from a bivariate $ t$-distribution with parameters $ \theta =2$, $ \rho =0,$ and $ \lambda ^{(3)}_U=0.1817$.

The empirical study shows that the TDC estimator $ \hat{\lambda}^{(3)}_U$ outperforms the other two estimators. For $ m=2000,$ the bias (RMSE) of $ \hat{\lambda}^{(1)}_U$ is three (two and a half) times larger than the bias (RMSE) of $ \hat{\lambda}^{(3)}_U$, whereas the bias (RMSE) of $ \hat{\lambda}^{(2)}_U$ is two (ten percent) times larger than the bias (RMSE) of $ \hat{\lambda}^{(1)}_U$. More empirical and statistical results regarding the estimators $ \hat{\lambda}^{(1)}_U$ and $ \hat{\lambda}^{(2)}_U$ are given in Schmidt and Stadtmüller (2003). However, note that the estimator $ \hat{\lambda}^{(3)}_U$ was especially developed for bivariate elliptically-contoured distributions. Thus, the estimator $ \hat{\lambda}^{(1)}_U$ is recommended for TDC estimations of non-elliptical or unknown bivariate distributions.