Estimating Familial Correlations Using a Kotz Type Density

<p>Two useful familial correlations often used to study the resemblance between the family members are the sib-sib correlation (ρ_<em>ss</em>) and the mom-sib or parent-sib correlation (ρ_<em>ps</em>). 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 ρ_<em>ss</em> and ρ<em>_ps </em>by minimizing the objective function,[special characters omitted]where Σ is a positive definite matrix with an appropriate structure involving ρ_<em>ss</em> and ρ_<em>ps</em>. 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 ρ_<em>ss</em>and ρ_<em>ps</em> 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.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics & Statistics

Year dc:date.available

2006

Author dc:creator

• Helu, Amal

Contributors dc:contributor

• Dayanand N. Naik
• N. Rao Chaganty
• Edward Markowski

• Familial
• Kotz-type density
• Statistics and Probability

Statement dc:rights

• <p>In Copyright. URI: <a href="http://rightsstatements.org/vocab/InC/1.0/">http://rightsstatements.org/vocab/InC/1.0/</a> This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).</p>

Identifier

9780542360909

OAI identifier oai:identifier

oai:digitalcommons.odu.edu:mathstat_etds-1018