Keywords

measure of concordance, copula

Abstract

A measure of concordance, $\kappa$, is of polynomial type if and only if $\kappa (tA+(1-t)B)$ is a polynomial in $t$ where $A$ and $B$ are 2-copulas. The degree of such a type of measure of concordance is simply the highest degree of the polynomial associated with $\kappa$. In previous work [2], [3], properties of measures of concordance preserving convex sums (equivalently measures of concordance of polynomial type degree one) were established; however, a characterization was not made. Here a characterization is made using approximations involving doubly stochastic matrices. Other representations are provided from this characterization leading naturally to two interpretations of degree one measures of concordance. The existence of a family of measures of concordance of polynomial type having higher degree generated by a certain family of Borel measures on $(0,1)^{2n}$ is also shown. The representation of this family immediately leads to a probabilistic interpretation for all finite measures in $d_n$. Also, higher degree analogs of commonly known degree one measures of concordance are given as examples. For the degree 2 case in particular, we see there is no finite measure in $d_2$ generating Kendall's tau. Finally, another family of measures of concordance is given containing those generated by finite measures in $d_2$ as well as Kendall's tau.

Notes

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Graduation Date

2004

Semester

Fall

Advisor

Taylor, Michael

Degree

Doctor of Philosophy (Ph.D.)

College

College of Arts and Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0000254

URL

http://purl.fcla.edu/fcla/etd/CFE0000254

Language

English

Release Date

January 2014

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

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Included in

Mathematics Commons

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