# HoeffdingD

HoeffdingD[v1,v2]

gives Hoeffding's dependence measure for the vectors v1 and v2.

HoeffdingD[m]

gives Hoeffding's dependence measure for the matrix m.

HoeffdingD[m1,m2]

gives Hoeffding's dependence measure for the matrices m1 and m2.

HoeffdingD[dist]

gives Hoeffding's matrix for the multivariate symbolic distribution dist.

HoeffdingD[dist,i,j]

gives the element of for the multivariate symbolic distribution dist.

# Details

• HoeffdingD[v1,v2] gives Hoeffding's dependence measure between v1 and v2.
• Hoeffding's is a measure of dependence based on the relative order of elements in the two lists.
• Hoeffding's between v1 and v2 is given by , where is the number of observations in v1, , , , for , is the rank of v1i, is the rank of v2i, and is equal to Boole[a<b].
• The arguments v1 and v2 can be any realvalued vectors of equal length greater than 5.
• For a matrix m with columns, HoeffdingD[m] is a × matrix of the dependence measures between columns of m.
• For an × matrix m1 and an × matrix m2, HoeffdingD[m1,m2] is a × matrix of the dependence measures between columns of m1 and columns of m2.
• HoeffdingD[dist,i,j] is given by 30 Expectation[(F[x,y]-G[x]H[y])^2,{x,y}disti,j], where F[x,y], G[x], and H[y] are the CDFs of the , , and marginals of dist respectively.
• HoeffdingD[dist] gives a matrix where the entry is given by HoeffdingD[dist,i,j].

# Examples

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## Basic Examples(4)

Hoeffding's for two vectors:

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Hoeffding's for a matrix:

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Hoeffding's for two matrices:

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Compute Hoeffding's for a bivariate distribution:

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Compare to a simulated value:

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