Copula-Based Dependence Measures For Piecewise Monotonicity
Abstract
The aim of the present paper is to develop and examine association coefficients which can be helpfully applied in the framework of regression analysis. The construction of the coeffiecients is connected with the well-known Spearman coeffiecient and extensions of it (see Liebscher [5]). The proposed coeffiecient measures the discrepancy between the data points and a function which is strictly increasing on one interval and strictly decreasing in the remaining domain.We prove statements about the asymptotic behaviour of the estimated coeffiecient (convergence rate, asymptotic normality).
-
[1] Brinkman, N. (1981). Ethanol fuel-A single-cylinder engine study of efficiency and exhaust emissions. SAE Technical Papers 810345. Available at http://dx.doi.org/10.4271/810345.
-
[2] Bücher, A., J. Segers, and S. Volgushev (2014). When uniform weak convergence fails: empirical processes for dependence functions and residuals via epi- and hypographs. Ann. Statist. 42(4), 1598-1634.
-
[3] Grothe, O., J. Schnieders, and J. Segers (2014). Measuring association and dependence between random vectors. J. Multivariate Anal. 123, 96-110.
-
[4] Juditsky, A. and A. Nemirovski (2002). On nonparametric tests of positivity/monotonicity/convexity. Ann. Statist. 30(2), 498-527.
-
[5] Liebscher, E. (2014). Copula-based dependence measures. Depend. Model. 2(1), 49-64.
-
[6] Nelsen, R. B. (2006). An Introduction to Copulas. Second Edition. Springer, New York.
-
[7] Pollard, D. (1984). Convergence of Stochastic Processes. Springer-Verlag, New York.
-
[8] Rockafellar, R. and R.-B. Wets (1998). Variational Analysis. Springer-Verlag, Berlin.
-
[9] Scarsini, M. (1984). On measures of concordance. Stochastica 8(3), 201-218.
-
[10] Schmid, F., R. Schmidt, T. Blumentritt, S. Gaißer, and M. Ruppert (2010). Copula-based measures of multivariate association. In P. Jaworski, F. Durante, W.K. Härdle, T. Rychlik (Eds.), Copula Theory and its Applications, pp. 209-236. Springer, Heidelberg.
-
[11] Serfling, R. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.
-
[12] Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229-231.
-
[13] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.
-
[14] Wang, J. C. and M. C. Meyer (2011). Testing the monotonicity or convexity of a function using regression splines. Canad. J. Statist. 39(1), 89-107.
