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I know what the properties "grounded" and "2-increasing" means in copula functions definition but actually I can't understand the reason behind these two! I mean why it is necessary for copulas to be grounded and 2-increasing?

Any help would be appreciated!

Infinity
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1 Answers1

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One interpretation of the requirements is from Sklar's theorem which says the joint cumulative of two random variables can be expressed as a Copula whose arguments are the marginal distributions of either random variable. Then the grounded condition makes sure you assign 0 probability when one of the marginals is 0. The 2 increasing is a generalization of the requirement that a one dimensional cdf is increasing. Here's a nice summary of all this.

Alex R.
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  • Dear Alex, thanks for your beneficial response. About the 2-increasing property, does this property belong to 2-dimension functions? As the dimension increases, for example to 5, is it correct to say this function is 5-increasing? Thanks in advance! – Infinity Apr 29 '15 at 20:40
  • See this answer: http://math.stackexchange.com/questions/568669/2-increasing-functions

    I think in general you want the gradient to point into the positive quadrant of your space. So a $n$ dimensional increasing function would have gradient in the regiion $(+,+,+,\cdots,+)$.

     – Alex R. Apr 29 '15 at 21:38