Considerable research has been done on measuring dependence between random variables. The correlation coefficient is the most widely studied linear measure of dependence. However, the limitation of linearity limits its application. The informational coefficient of correlation is defined in terms of mutual information. It also has some deficiencies, such as it is only normalized to continuous random variables. Based on the concept of the informational coefficient of correlation, a new dependence measure, which we call the L-measure, is proposed in this work which generalizes Linfoot's measure for both continuous and discrete random variables. To further elucidate its properties, simulated models are used, and estimation algorithms are also discussed. Furthermore, another measure based on the L-measure, which we call the intrinsic L-measure, is studied for the purpose of studying nonlinear dependence. Based on criteria for a dependence measure presented by Renyi and simulation results in this thesis, we believe that the L-measure is satisfactory as a dependence measure. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-09-30 14:29:35.153

In epidemiological studies, subjects with disease (prevalent cases) differ from newly diseased (incident cases). They tend to survive longer due to sampling bias, and related covariates will also be biased. Methods for regression analyses have recently been proposed to measure the potential effects of covariates on survival. The goal is to extend the dependence measure of Kent (1983), based on the information gain, in the context of length-biased sampling. In this regard, to estimate information gain and dependence measure for length-biased data, we propose two different methods namely kernel density estimation with a regression procedure and parametric copulas. We will assess the consistency for all proposed estimators. Algorithms detailing how to generate length-biased data, using kernel density estimation with regression procedure and parametric copulas approaches, are given. Finally, the performances of the estimated information gain and dependence measure, under length-biased sampling, are demonstrated through simulation studies.