Answer-first summary for fast verification
Answer: I and IV only
## Explanation Let's analyze each statement: **Statement I: Correct** - Copula functions allow us to model the dependence structure separately from the marginal distributions. This is a key advantage of copulas - they decouple the dependence structure from the marginal distributions. **Statement II: Incorrect** - Transformation of variables can change their correlation structure. For example, nonlinear transformations can alter the correlation between variables. **Statement III: Incorrect** - Correlation requires that the variables have finite variances. If a distribution doesn't have a defined variance (like some heavy-tailed distributions), correlation is not well-defined and cannot be a useful measure. **Statement IV: Correct** - Correlation is indeed a good measure of dependence for multivariate elliptical distributions (such as the multivariate normal distribution), where it captures the complete dependence structure. Therefore, only statements I and IV are correct, making option A the correct answer.
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Which of the following statements about correlation and copula are correct?
I. Copula enables the structures of correlation between variables to be calculated separately from their marginal distributions.
II. Transformation of variables does not change their correlation structure.
III. Correlation can be a useful measure of the relationship between variables drawn from a distribution without a defined variance.
IV. Correlation is a good measure of dependence when the measured variables are distributed as multivariate elliptical.
A
I and IV only
B
II, III, and IV only
C
I and III only
D
II and IV only
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