A full ranking of n items is simply an ordering of all these items, of the form: first choice, second choice, โ{128}ข. . , n-th choice. If two judges each rank the same n items, statisticians have used various metrics to measure the closeness of the two rankings, including Kenยญ dall's tau, Spearman's rho, Spearman's footrule, Ulam's metric, Hal1l11ing distance, and Cayley distance. These metrics have been emยญ ployed in many contexts, in many applied statistical and scientific problems. Thi s monograph presents genera 1 methods for extendi ng these metri cs to partially ranked data. Here "partially ranked data" refers, for instance, to the situation in which there are n distinct items, but each judge specifies only his first through k-th choices, where k < n. More complex types of partially ranked data are also investigated. Group theory is an important tool for extending the metrics. Full rankings are identified with elements of the permutation group, whereas partial rankings are identified with points in a coset space of the permutation group. The problem thus becomes one of exยญ tending metrics on the permutation group to metrics on a coset space of the permutation group. To carry out the extens"ions, two novel methods -- the so-called Hausdorff and fixed vector methods -- are introduced and implemented, which exploit this group-theoretic structure. Various data-analytic applications of metrics on fully ranked data have been presented in the statistical literature
CONTENT
I. Introduction and Outline -- II. Metrics on Fully Ranked Data -- A. Permutations: Some Important Conventions -- B. Metrics on Permutations: Discussion and Exampl es -- C. The Requirement of Rightโ{128}{148}Invariance -- III. Metrics on Partially Ranked Data: The Case where Each Judge Lists His k Favorite Items Out of n -- A. The Coset Space Sn/Sn-k -- B. The Hausdorff Metrics on Sn/Sn-k -- C. The Fixed Vector Metrics on Sn/Sn-k -- IV. Metrics on Other Types of Partially Ranked Data -- A. The Coset Space Sn/S, Where S = Sn1 ร{151}Sn2 ร{151} โ{128}ฆร{151}Snr -- B. The Hausdorff Metrics on Sn/S -- C. The Fixed Vector Metrics on Sn/S -- D. Hausdorff Distances between Different Types of Partially Ranked Data: A Complete Proof of the Main Theorem -- E. The Tied Ranks Approach to Metrizing Partially Ranked Data -- V. Distributional Properties of the Metrics -- A. Exact Distributions -- B. Asymptotic Distributions -- VI. Data Analysis, Using the Metrics -- A. Fitting Probability Models to Partially Ranked Data -- B. Multidimensional Scaling for Partially Ranked Data -- C. Two Sample Problems for Partially Ranked Data -- Appendix A โ{128}{148} The Existence Of Fixed Vectors -- Appendix C โ{128}{148} Fortran Subroutines For Fitting Mallowsโ{128}{153} Model To Partially Ranked Data -- Appendix E โ{128}{148} Comparison Of Exact And Asymptotic Distributions -- Index Of Notation
