Date of Award
Doctor of Philosophy (PhD)
Mathematics and Statistics
Computational and Applied Mathematics
Dayanand N. Naik
N. Rao Chaganty
Two useful familial correlations often used to study the resemblance between the family members are the sib-sib correlation (ρss) and the mom-sib or parent-sib correlation (ρps). Since their introduction early in the last century by Galton, Fisher and others, many improved estimators of these correlations have been suggested in the literature. Several moment based estimators as well as the maximum likelihood estimators under the assumption of multivariate normality have been extensively studied and compared by various authors. However, the performance of these estimators when the data are not from multivariate normal distribution is poor. In this dissertation, we provide alternative estimates of ρss and ρps by minimizing the objective function,[special characters omitted]where Σ is a positive definite matrix with an appropriate structure involving ρss and ρps. Using extensive simulations from different multivariate distributions and using the bias, the mean squared error, and Pitman probability of nearness we have established that the alternative estimators are better than the existing estimators in most situations. The problems of testing of hypothesis about ρss and ρps and those of testing the equality of two sib-sib correlations and two mom-sib correlations are also considered. Alternative tests using Srivastava's well known estimators of sib-sib and mom-sib correlations and their asymptotic variances are proposed and compared using simulations. The proposed tests have better estimated sizes and powers than the likelihood based tests when data are from a multivariate normal distribution. Proposed methods are illustrated on Galton's famous classical data set on statures of families. These data are important, in that, the original note book on which these data were recorded by Galton in 1886 has been recently discovered and digitalized.
"Estimating Familial Correlations Using a Kotz Type Density"
(2006). Doctor of Philosophy (PhD), Dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/scxt-a694