Date of Award
1990
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Abstract
Algorithms for enumerating the exact null distributions of Kendall's S and Spearman's D statistics, when there are ties in one or both of the rankings, are presented. An expression, which is used to provide a simple proof of the asymptotic normality of the score S when both rankings are tied, is obtained for the cumulant generating function of S. The usefulness of an Edgeworth approximation to the null distribution of S in the general case of tied rankings is investigated and compared with the standard normal approximation.;Exact and asymptotic results are developed for the distribution of Kendall's partial rank correlation statistic {dollar}t\sb{lcub}12.3{rcub}{dollar}, under the complete null hypothesis. A probability model, with the property that for the associated permutations E(t) = {dollar}\tau{dollar}, is developed for the elements of an inversion vector. The variance of t under this probability model is derived, an application of this result to hypothesis testing is presented, and an algorithm for simulating rankings of size n, so that E(t) = {dollar}\tau{dollar}, is given.;An asymptotic variance estimator for {dollar}t\sb{lcub}12.3{rcub}{dollar} is derived and the asymptotic normality of {dollar}t\sb{lcub}12.3{rcub}{dollar}, under {dollar}H\sb o{dollar}: {dollar}\tau\sb{lcub}12.3{rcub}{dollar} = 0 and for the general case of variates with underlying parental correlation, is established. Monte Carlo simulation is used to show that when the magnitudes of {dollar}t\sb{lcub}13{rcub}{dollar} and {dollar}t\sb{lcub}23{rcub}{dollar} are both moderately large, {dollar}t\sb{lcub}12.3{rcub}{dollar} is not a suitable statistic for testing the hypothesis {dollar}H\sbsp{lcub}o{rcub}{lcub}\prime{rcub}{dollar}: {dollar}X\sb1{dollar} and {dollar}X\sb2{dollar} are conditionally, given {dollar}X\sb3{dollar}, independent of each other. Consequently, a simulation study of partial Spearman's {dollar}\rho{dollar} is implemented. This study shows that {dollar}r\sb{lcub}s,12.3{rcub}{dollar}, when corrected for bias in {dollar}r\sb{lcub}s,12{rcub}{dollar} etc., provides a satisfactory test statistic whose asymptotic distribution under {dollar}H\sb o{dollar}: {dollar}\rho\sb{lcub}\rm s,12{rcub}{dollar} = 0 may be adequately approximated by its asymptotic distribution under the complete null hypothesis.
Recommended Citation
Valz, Paul Dexter, "Developments In Rank Correlation Procedures For Trend Detection In The Analysis Of Water Quality Parameters" (1990). Digitized Theses. 1909.
https://ir.lib.uwo.ca/digitizedtheses/1909