Interview Question: Nonlinear Dependence

Sample Question #118 (probability theory)

If two random variables are linearly independent of each other but exhibit nonlinear dependence, how do we measure such nonlinear dependence? Can you explain your answer in some detail?

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One Response to Interview Question: Nonlinear Dependence

  1. Brett says:

    ANSWER
     
    The answer to this esoteric question is nonlinear dependence can be modeled using copula functions. Copula functions are based on the Theorem of Sklar, which states that given any join cdf F(x1, x2, …, xN), there is a continuous function C such that F(x1,…xN) = C(G_1(x1), …, G_N(xN)), where G are the marginal cdf’s. The copula function C thus maps the functional relationship between the random variables into a functional relationship between their marginal distributions. This functional relationship is usually nonlinear, thus allowing us to derive nonlinear inter-variable statistics such as nonlinear covariances and correlations.
     

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