Sensitivity of Linear Models with Respect to the Covariance Matrix

1985 ◽  
pp. 223-230
Author(s):  
E. Neuwirth
Keyword(s):  
Covariance Matrix ◽  
Linear Models

Covariance matrix of the bias-corrected maximum likelihood estimator in generalized linear models

2013 ◽  
Vol 55 (3) ◽  
pp. 643-652
Author(s):  
Gauss M. Cordeiro ◽  
Denise A. Botter ◽  
Alexsandro B. Cavalcanti ◽  
Lúcia P. Barroso
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Covariance Matrix ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Likelihood Estimator

scholarly journals Best linear unbiased estimation for varying probability with and without replacement sampling

2019 ◽  
Vol 7 (1) ◽  
pp. 78-91
Author(s):  
Stephen Haslett
Keyword(s):  
Parameter Estimation ◽  
Covariance Matrix ◽  
Linear Models ◽  
Unbiased Estimation ◽  
Error Covariance Matrix ◽  
Best Linear Unbiased Estimation ◽  
Error Covariance ◽  
Inclusion Probabilities ◽  
Best Linear Unbiased ◽  
Linear Unbiased Estimation

Abstract When sample survey data with complex design (stratification, clustering, unequal selection or inclusion probabilities, and weighting) are used for linear models, estimation of model parameters and their covariance matrices becomes complicated. Standard fitting techniques for sample surveys either model conditional on survey design variables, or use only design weights based on inclusion probabilities essentially assuming zero error covariance between all pairs of population elements. Design properties that link two units are not used. However, if population error structure is correlated, an unbiased estimate of the linear model error covariance matrix for the sample is needed for efficient parameter estimation. By making simultaneous use of sampling structure and design-unbiased estimates of the population error covariance matrix, the paper develops best linear unbiased estimation (BLUE) type extensions to standard design-based and joint design and model based estimation methods for linear models. The analysis covers both with and without replacement sample designs. It recognises that estimation for with replacement designs requires generalized inverses when any unit is selected more than once. This and the use of Hadamard products to link sampling and population error covariance matrix properties are central topics of the paper. Model-based linear model parameter estimation is also discussed.

ANOVA models with random effects: an approach via symmetry

1986 ◽  
Vol 23 (A) ◽  
pp. 355-368 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Covariance Matrix ◽  
Random Effects ◽  
Spectral Decomposition ◽  
Linear Models ◽  
Random Variables ◽  
Components Of Variance

The standard ANOVA models with random effects for multi-indexed arrays of random variables with an arbitrary nesting structure on the indices are considered from the viewpoint of symmetry. It is found that the covariance matrix of such an array has sufficient symmetry to permit viewing the usual components of variance as a generalised spectrum and the linear models of random effects as a generalised spectral decomposition.

scholarly journals Linear Models with Response Functions Based on the Laplace Distribution: Statistical Formulae and Their Application to Epigenomics

2013 ◽  
Vol 2013 ◽  
pp. 1-22
Author(s):  
C. Z. W. Hassell Sweatman ◽  
G. C. Wake ◽  
A. B. Pleasants ◽  
C. A. McLean ◽  
A. M. Sheppard
Keyword(s):  
Linear Model ◽  
Covariance Matrix ◽  
Response Function ◽  
Linear Models ◽  
Likelihood Ratio Statistic ◽  
Laplace Distribution ◽  
Simple Expression ◽  
Modulus Function ◽  
Response Functions ◽  
Statistical Application

The statistical application considered here arose in epigenomics, linking the DNA methylation proportions measured at specific genomic sites to characteristics such as phenotype or birth order. It was found that the distribution of errors in the proportions of chemical modification (methylation) on DNA, measured at CpG sites, may be successfully modelled by a Laplace distribution which is perturbed by a Hermite polynomial. We use a linear model with such a response function. Hence, the response function is known, or assumed well estimated, but fails to be differentiable in the classical sense due to the modulus function. Our problem was to estimate coefficients for the linear model and the corresponding covariance matrix and to compare models with varying numbers of coefficients. The linear model coefficients may be found using the (derivative-free) simplex method, as in quantile regression. However, this theory does not yield a simple expression for the covariance matrix of the coefficients of the linear model. Assuming response functions which are 𝒞2 except where the modulus function attains zero, we derive simple formulae for the covariance matrix and a log-likelihood ratio statistic, using generalized calculus. These original formulae enable a generalized analysis of variance and further model comparisons.

Covariance Matrix Formula for Generalized Linear Models with Unknown Dispersion

2006 ◽  
Vol 35 (1) ◽  
pp. 113-120 ◽  
Author(s):  
Gauss M. Cordeiro ◽  
Lúcia P. Barroso ◽  
Denise A. Botter
Keyword(s):  
Covariance Matrix ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Matrix Formula
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