An ordering of dependence. (English) Zbl 0768.62045
Topics in statistical dependence, Proc. Symp. Depend. Stat. Probab., Somerset/PA (USA) 1987, IMS Lect. Notes, Monogr. Ser. 16, 403-414 (1990).
Summary: [For the entire collection see Zbl 0760.00004.]
An ordering of dependence is defined on the space of probability measures on a finite product space, with fixed marginals. The definition of this ordering involves the Lorenz curve of the likelihood ratio of a probability measure w.r.t. the product measure of the marginals. A minimal element w.r.t. this dependence ordering always exists and equals the product measure. Conditions for the existence of a maximal element are examined. The ordering is generalized to the case of infinite spaces. Comparison with some other orders of dependence is considered.
An ordering of dependence is defined on the space of probability measures on a finite product space, with fixed marginals. The definition of this ordering involves the Lorenz curve of the likelihood ratio of a probability measure w.r.t. the product measure of the marginals. A minimal element w.r.t. this dependence ordering always exists and equals the product measure. Conditions for the existence of a maximal element are examined. The ordering is generalized to the case of infinite spaces. Comparison with some other orders of dependence is considered.
MSC:
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
