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Title: Covariance modelling with hypersphere decomposition method and modified hypersphere decomposition method
Author: Li, Qingze
Awarding Body: University of Manchester
Current Institution: University of Manchester
Date of Award: 2018
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In statistical studies, covariance estimation is of great importance to the accuracy of estimators of mean parameters and statistical testing. The main problems for covariance estimation are high-dimensionality, symmetry, and positive-definiteness. High-dimensional situations, where the number of repeated measurements p is less than the sample size n, are usually due to the high cost of each repeated measurement, such as gene test and blood tests for HIV. Under the high-dimensional situation, sample covariance matrices are singular, which are not appropriate for obtaining the precision matrices and statistical tests. A specific group/family of covariance structures usually be pre-assumed to reduce the dimension of unknown parameters. While there is not any test yet to identify if the optimal structure is pre-selected. Unconstrained Parametrizations for variance-covariance matrices with different decomposition methods are commonly used in covariance modelling studies. Popular methods include Modify Cholesky Decomposition (MCD), Alternative Cholesky Decomposition (ACD). These data-driving methods can select the optimal covariance model by non-parametric analysis and model selection study. But the statistical interpretations of those two methods are not straightforward. In recent, a new proposed decomposition method, Hypersphere Decomposition (HPC), is applied in covariance modelling to improve the interpretation problem. While all three methods above, MCD, ACD and HPC are order-dependent due to the definition of their decomposition. That makes these methods not appropriate for analysing data with no natural order. In this paper, we propose a new method, Modified Hypersphere Decomposition (MHPC), by redefining the angular translation in HPC. This new method has most advantages of HPC, meanwhile is order-independent. Though the resulting estimator of mean parameters of Generalized Estimating Equations (GEE) is robust to the structure of working covariance matrices and distribution assumptions, by estimating mean and covariance matrices jointly in GEE can get an unconstrained data-driving covariance estimator and improve the efficiency of the estimator of mean parameters. Ye and Pan (2006) apply MCD method with GEE to model the mean and covariance jointly. Here we apply HPC and our MHPC with GEE, which can provide a better statistical interpretation of the estimator of covariance model with little cost on the efficiency of mean parameter estimator, comparing to MCD GEE method. The widely applied Growth Curve Model (GCM) in longitudinal studies requires an appropriate estimation of within-subject covariances to produce an efficient estimator of the mean parameters. In this paper, we apply a recently introduced data-driving method, Hypersphere Decomposition (HPC), on the modelling of the within-group covariance matrices in Growth Curve Model. And we further study the asymptotic properties of the estimators under different dimensional situations.
Supervisor: Pan, Jianxin ; Donev, Alexander Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Covariance Modelling ; Hypersphere Decomposition ; Modified Hypersphere Decomposition ; Growth Curve Model