On equality and proportionality of ordinary least squares, weighted least squares and best linear unbiased estimators in the general linear model

Examination of skewness makes academics more aware of the importance of accurate statistical analysis. Undoubtedly, most phenomena contain a certain percentage of skewness which resulted to the appearance of what is -called "asymmetry" and, consequently, the importance of the skew normal family . The epsilon skew normal distribution ESN (μ, σ, ε) is one of the probability distributions which provide a more flexible model because the skewness parameter provides the possibility to fluctuate from normal to skewed distribution. Theoretically, the estimation of linear regression model parameters, with an average error value that is not zero, is considered a major challenge due to having difficulties, as no explicit formula to calculate these estimates can be obtained. Practically, values for these estimates can be obtained only by referring to numerical methods. This research paper is dedicated to estimate parameters of the Epsilon Skew Normal General Linear Model (ESNGLM) using an adaptive least squares method, as along with the employment of the ordinary least squares method for estimating parameters of the General Linear Model (GLM). In addition, the coefficient of determination was used as a criterion to compare the models’ preference. These methods were applied to real data represented by dollar exchange rates. The Matlab software was applied in this work and the results showed that the ESNGLM represents a satisfactory model.

Download Full-text

The use of MIXED models in the analysis of animal experiments with repeated measures data

Canadian Journal of Animal Science ◽

10.4141/a03-123 ◽

2004 ◽

Vol 84 (1) ◽

pp. 1-11 ◽

Cited By ~ 312

Author(s):

Z. Wang and L. A. Goonewardene

Keyword(s):

Least Squares ◽

Linear Model ◽

Repeated Measures ◽

Mixed Model ◽

Covariance Structure ◽

General Linear Model ◽

Ordinary Least Squares ◽

General Linear ◽

Repeated Measures Data ◽

Split Plot

The analysis of data containing repeated observations measured on animals (experimental unit) allocated to different treatments over time is a common design in animal science. Conventionally, repeated measures data were either analyzed as a univariate (split-plot in time) or a multivariate ANOVA (analysis of contrasts), both being handled by the General Linear Model procedure of SAS. In recent times, the mixed model has become more appealing for analyzing repeated data. The objective of this paper is to provide a background understanding of mixed model methodology in a repeated measures analysis and to use balanced steer data from a growth study to illustrate the use of PROC MIXED in the SAS system using five covariance structures. The split-plot in time approach assumes a constant variance and equal correlations (covariance) between repeated measures or compound symmetry, regardless of their proximity in time, and often these assumptions are not true. Recognizing this limitation, the analysis of contrasts was proposed. If there are missing measurements, or some of the data are measured at different times, such data were excluded resulting in inadequate data for a meaningful analysis. The mixed model uses the generalized least squares method, which is generally better than the ordinary least squares used by GLM, if the appropriate covariance structure is adopted. The presence of unequally spaced and/or missing data does not pose a problem for the mixed model. In the example analyzed, the first order ante dependence [ANTE(1)] covariance model had the lowest value for the Akaike and Schwarz’s Bayesian information criteria fit statistics and is therefore the model that provided the best fit to our data. Hence, F values, least square estimates and standard errors based on the ANTE (1) were considered the most appropriate from among the five models demonstrated. It is recommended that the mixed model be used for the analysis of repeated measures designs in animal studies. Key words: Repeated measures, General Linear Model, Mixed Model, split-plot, covariance structure

Download Full-text

On equality of ordinary least squares estimator, best linear unbiased estimator and best linear unbiased predictor in the general linear model

Journal of Statistical Planning and Inference ◽

10.1016/j.jspi.2008.08.015 ◽

2009 ◽

Vol 139 (4) ◽

pp. 1522-1529 ◽

Cited By ~ 7

Author(s):

Yonghui Liu

Keyword(s):

Least Squares ◽

Linear Model ◽

Unbiased Estimator ◽

General Linear Model ◽

Ordinary Least Squares ◽

Least Squares Estimator ◽

Best Linear Unbiased Estimator ◽

Best Linear Unbiased Predictor ◽

Ordinary Least Squares Estimator ◽

Best Linear Unbiased

Download Full-text

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

Open Mathematics ◽

10.1515/math-2017-0013 ◽

2017 ◽

Vol 15 (1) ◽

pp. 126-150 ◽

Cited By ~ 10

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.

Download Full-text