By Svetlozar T. Rachev, Stoyan V. Stoyanov, Visit Amazon's Frank J. Fabozzi Page, search results, Learn about Author Central, Frank J. Fabozzi,
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Extra resources for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures
6. 4) reveals that, if the random variable Y has independent components, then the density of the corresponding copula, denoted by c0 , is a constant in the unit hypercube, c0 (u1 , . . , un ) = 1 and the copula C0 has the following simple form, C0 (u1 , . . , un ) = u1 . . un . This copula characterizes stochastic independence. Now let us consider a density c of some copula C. 4) is a ratio of two positive quantities because the density function can only take nonnegative values. For each value of the vector of arguments y = (y1 , .
Ng (1994). Symmetric multivariate and related distributions, New York: Marcel Dekker. Johnson, N. , S. Kotz and A. W. Kemp (1993). , New York: John Wiley & Sons. Larsen, R. , and M. L. Marx (1986). An introduction to mathematical statistics and its applications, Englewed Clifs, NJ: Prentice Hall. Mikosch, T. (2006). ‘‘Copulas—tales and facts,’’ Extremes 9: 3–20. Patton, A. J. (2002). Application of copular theory in financial econometrics, Doctoral Dissertation, Economics, University of California, San Diego.
In this case, it has a simpler form because the marginal distributions are uniform. The lower and the upper Fr´echet bounds equal W(u1 , . . , un ) = max(u1 + · · · + un + 1 − n, 0) and M(u1 , . . , un ) = min(u1 , . . , un ) respectively. Fr´echet-Hoeffding inequality is given by W(u1 , . . , un ) ≤ C(u1 , . . , un ) ≤ M(u1 , . . , un ). In the two-dimensional case, the inequality reduces to max(u1 + u2 − 1, 0) ≤ C(u1 , u2 ) ≤ min(u1 , u2 ). In the two-dimensional case only, the lower Fr´echet bound, sometimes referred to as the minimal copula, represents perfect negative dependence between the two random variables.
Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures by Svetlozar T. Rachev, Stoyan V. Stoyanov, Visit Amazon's Frank J. Fabozzi Page, search results, Learn about Author Central, Frank J. Fabozzi,