Implement the study in Example 14.3 and compare your results to the example. Example 14.3. In this

Implement the study in Example 14.3 and compare your results to the example.

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Example 14.3. In this example we illustrate how to isolate the nonlinear dependence between random vectors to test for nonlinearity. Gumbel's bivariate exponential distribution [Gumbel, 1961] has density function f(x, y; 0) = [(1 + 0x)(1+0y)] exp(-x - y - Oxy), x,y>0;0 0 1. The marginal distributions are standard exponential, so there is a strong non- linear, but monotone dependence relation between X and Y. The conditional density is f(y|x) = e(+0x)y [(1 + 0x) (1 + 0y) 0], y > 0. If = 0, then fx,y (x, y) = fx (1) fy (y) and independence holds, so p = 0. At the opposite extreme if 0 = 1, then p= -0.40365 (see [Kotz et al., 2000, Sec. 2.2]). Simulated data was generated using the conditional distribution function approach outlined in Johnson [1987]. Empirical power of dCov and correlation tests for the case = 0.5 are compared in Figure 14.3(a), estimated from 10,000 test decisions each for sample sizes {10:100(10), 120:200(20), 250, 300}. This comparison reveals that the correlation test is more powerful than dCov against this alternative, which is not unexpected because E[Y|X = x] = (1+0+x0)/(1+ x) is monotone. power 19 0.8 0.6 8 0.2 0 .-.- dCov Pearson rank-dCov Spearman T 50 T 100 T 150 sample size (a) T 200 T 250 300 0.8 0.6 0.2 8 0 ---- dCov Pearson rank-dCov Spearman 50 T 100 T 150 sample size (b) T 200 T 250 300 While we cannot split the dCor or dCov coefficient into linear and nonlinear components, we can extract correlation first and then compute dCor on the residuals. In this way one can separately analyze the linear and nonlinear components of bivariate or multivariate dependence relations. Xif To extract the linear component of dependence, fit a linear model Y = XB + to the sample (X, Y) by ordinary least squares. It is not necessary to test whether the linear relation is significant. The residuals ; = X3 - Y; are uncorrelated with the predictors X. Apply the dCov test of independence to (X, ). Returning to the Gumbel bivariate exponential example, we have extracted the linear component and applied dCov to the residuals of a simple linear re- gression model. Repeating the power comparison described above on (X, 2) data, we obtained the power estimates shown in Figure 14.3(b). One can note that power of dCov tests is increasing to 1 with sample size, exhibiting statis- tical consistency against the nonlinear dependence remaining in the residuals of the linear model. This procedure is easily applied in arbitrary dimension. One can fit a linear multiple regression model or a model with multivariate response to extract the linear component of dependence. This has important practical applications for evaluating models in higher dimension.