## COVARIANCE MODELLING WITH HYPERSPHERE DECOMPOSITION METHOD AND MODIFIED HYPERSPHERE DECOMPOSITION METHOD

UoM administered thesis: Phd

• Authors:
• Qingze Li

## Abstract

In statistical studies, covariance estimation is of great importance to the accu- racy 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 measure- ment, 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 struc- tures 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. Uncon- strained Parametrizations for variance-covariance matrices with different decomposi- tion methods are commonly used in covariance modelling studies. Popular methods include Modify Cholesky Decomposition (MCD), Alternative Cholesky Decomposi- tion (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 decomposi- tion 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 transla- tion in HPC. This new method has most advantages of HPC, meanwhile is order- independent. Though the resulting estimator of mean parameters of Generalized Estimating Equa- tions (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 es- timator 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 covari- ance 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.

## Details

Original language English The University of Manchester Jianxin Pan (Supervisor)Alexander Donev (Supervisor) 1 Aug 2018