scholarly journals On best linear unbiased estimation and prediction under a constrained linear random-effects model

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bo Jiang ◽  
Yongge Tian
Keyword(s):  
Random Effects ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Covariance Matrices ◽  
Unbiased Estimation ◽  
Random Effects Model ◽  
Unknown Parameters ◽  
Best Linear Unbiased ◽  
Best Linear Unbiased Estimators ◽  
Linear Restrictions

<p style='text-indent:20px;'>This paper is concerned with solving some fundamental estimation, prediction, and inference problems on a linear random-effects model with its parameter vector satisfying certain exact linear restrictions. Our work includes deriving analytical formulas for calculating the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of all unknown parameters in the model by way of solving certain constrained quadratic matrix optimization problems, characterizing various mathematical and statistical properties of the predictors and estimators, establishing various fundamental rank and inertia formulas associated with the covariance matrices of predictors and estimators, and presenting necessary and sufficient conditions for several equalities and inequalities of covariance matrices of the predictors and estimators to hold.</p>

scholarly journals Some remarks on a pair of seemingly unrelated regression models

2019 ◽  
Vol 17 (1) ◽  
pp. 979-989 ◽  
Author(s):  
Jian Hou ◽  
Yong Zhao
Keyword(s):  
Linear Regression ◽  
Regression Models ◽  
Sufficient Conditions ◽  
Seemingly Unrelated Regression ◽  
Regression Equations ◽  
Linear Regression Models ◽  
Unknown Parameters ◽  
Unrelated Regression ◽  
Best Linear Unbiased ◽  
Best Linear Unbiased Estimators

Abstract Linear regression models are foundation of current statistical theory and have been a prominent object of study in statistical data analysis and inference. A special class of linear regression models is called the seemingly unrelated regression models (SURMs) which allow correlated observations between different regression equations. In this article, we present a general approach to SURMs under some general assumptions, including establishing closed-form expressions of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of all unknown parameters in the models, establishing necessary and sufficient conditions for a family of equalities of the predictors and estimators under the single models and the combined model to hold. Some fundamental and valuable properties of the BLUPs and BLUEs under the SURM are also presented.

scholarly journals Matrix rank and inertia formulas in the analysis of general linear models

2017 ◽  
Vol 15 (1) ◽  
pp. 126-150 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Statistical Analysis ◽  
Least Squares ◽  
Linear Models ◽  
Linear Regression Analysis ◽  
Ordinary Least Squares ◽  
Covariance Matrices ◽  
General Linear ◽  
General Linear Models ◽  
Best Linear Unbiased ◽  
Best Linear Unbiased Estimators

Abstract Matrix mathematics provides a powerful tool set for addressing statistical problems, in particular, the theory of matrix ranks and inertias has been developed as effective methodology of simplifying various complicated matrix expressions, and establishing equalities and inequalities occurred in statistical analysis. This paper describes how to establish exact formulas for calculating ranks and inertias of covariances of predictors and estimators of parameter spaces in general linear models (GLMs), and how to use the formulas in statistical analysis of GLMs. We first derive analytical expressions of best linear unbiased predictors/best linear unbiased estimators (BLUPs/BLUEs) of all unknown parameters in the model by solving a constrained quadratic matrix-valued function optimization problem, and present some well-known results on ordinary least-squares predictors/ordinary least-squares estimators (OLSPs/OLSEs). We then establish some fundamental rank and inertia formulas for covariance matrices related to BLUPs/BLUEs and OLSPs/OLSEs, and use the formulas to characterize a variety of equalities and inequalities for covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. As applications, we use these equalities and inequalities in the comparison of the covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. The work on the formulations of BLUPs/BLUEs and OLSPs/OLSEs, and their covariance matrices under GLMs provides direct access, as a standard example, to a very simple algebraic treatment of predictors and estimators in linear regression analysis, which leads a deep insight into the linear nature of GLMs and gives an efficient way of summarizing the results.

Theory of best integer equivariant estimation for contaminated normal and multivariate t-distribution with applications

2021 ◽  
Author(s):  
Peter Teunissen
Keyword(s):  
Normal Distribution ◽  
Least Squares ◽  
Unbiased Estimation ◽  
Multivariate T Distribution ◽  
Integer Least Squares ◽  
Wide Range ◽  
Best Linear Unbiased ◽  
Best Linear Unbiased Estimators ◽  
T Distribution ◽  
Multivariate T

&lt;p&gt;Best integer equivariant (BIE) estimators provide minimum mean squared error (MMSE) solutions to the problem of GNSS carrier-phase ambiguity resolution for a wide range of distributions. The associated BIE estimators are universally optimal in the sense that they have an accuracy which is never poorer than that of any integer estimator and any linear unbiased estimator. Their accuracy is therefore always better or the same as that of Integer Least-Squares (ILS) estimators and Best Linear Unbiased Estimators (BLUEs).&lt;/p&gt;&lt;p&gt;Current theory is based on using BIE for the multivariate normal distribution. In this contribution this will be generalized to the contaminated normal distribution and the multivariate t-distribution, both of which have heavier tails than the normal. Their computational formulae are presented and discussed in relation to that of the normal distribution. In addition a GNSS real-data based analysis is carried out to demonstrate the universal MMSE properties of the BIE estimators for GNSS-baselines and associated parameters.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;Keywords: &lt;/strong&gt;Integer equivariant (IE) estimation &amp;#183; Best integer equivariant (BIE) &amp;#183; Integer Least-Squares (ILS) . Best linear unbiased estimation (BLUE) &amp;#183; Multivariate contaminated normal &amp;#183; Multivariate t-distribution . Global Navigation Satellite Systems (GNSSs)&lt;/p&gt;

A matrix handling of predictions of new observations under a general random-effects model

2015 ◽  
Vol 29 ◽  
pp. 30-45 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Random Effects ◽  
Optimization Problem ◽  
General Assumption ◽  
Partial Ordering ◽  
General Linear ◽  
Random Effects Model ◽  
Best Linear Unbiased ◽  
Original Observation ◽  
Observation Vector ◽  
Analytical Expressions

Assume that a general linear random-effects model $\by = \bX\bbe + \bve$ is given, and new observations in the future follow the linear model $\by_{\!f} = \bX_{\!f}\bbe + \bve_{\!f}$. This paper shows how to establish all possible best linear unbiased predictors (BLUPs) under the general linear random-effects model with original and new observations from the original observation vector $\by$ under a most general assumption on the covariance matrix among the random vectors $\bbe$, $\bve$ and $\bve_{\!f}$. It utilizes a standard method of solving optimization problem in the L\"owner partial ordering on a constrained quadratic matrix-valued function, and obtains analytical expressions of the BLUPs, including those for $\by_{\!f}$, $\bX_{\!f}\bbe$ and $\bve_{\!f}$. In particular, some fundamental equalities for the BLUPs are established under the linear random-effects model.

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