A note on the unified theory of least squares

The general form of a matrix which appears in the normal equation for estimating parameters in the Gauss-Markoff linear model has been obtained.     

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

The equality of ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) in the general linear model with new observations is investigated through matrix rank method, some new necessary and sufficient conditions are given.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

On O.L.S. estimation in singular linear models with general positive definite dispersion matrix

A condition in Graybill [1976] for the O.L.S.estimator to be B.L.U.E. in a linear model with positive definite dispersion not necessarily proportional to the identity matrix is extended to cover the case of a singular linear model.     

A method for screening active effects in supersaturated designs

A supersaturated design (SSD) is a design whose run size is not enough for estimating all the main effects. The goal in conducting such a design is to identify, presumably only a few, relatively dominant active effects with a cost as low as possible. However, data analysis of such designs remains primitive: traditional approaches are not appropriate in such a situation and several methods which were proposed in the literature in recent years are effective when used to analyze two-level SSDs. In this paper, we introduce a variable selection procedure, called the PLSVS method, to screen active effects in mixed-level SSDs based on the variable importance in projection which is an important concept in the partial least-squares regression. Simulation studies show that this procedure is effective.     

A Jackknife Method for Estimation of Variance Components

This paper concerns a method of estimation of variance components in a random effect linear model. It is mainly a resampling method and relies on the Jackknife principle. The derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimator. Our resampling method is illustrated by an example given by C. R. Rao [7] and some optimal properties of our estimator are derived for this example. In the last part, this method is used to derive an estimation of variance components in a random effect linear model when one of the components is assumed to be known.     

Robust estimation in certain heteroscedastic linear models when there are many parameters

We study estimation of regression parameters in heteroscedastic linear models when the number of parameters is large. The results generalize work of Huber (1973), Yohai and Maronna (1979), and Carroll and Rupert (1982a).     

Methods for calculating stationary distribution in linear models of time series

The article deals with methods for computing the stationary marginal distribution in linear models of time series. Two approaches are described. First, an algorithm based on approximation of solution of the corresponding integral equation is briefly reviewed. Then, we study the limit behaviour of the partial sums c ₁ η₁+c ₂ η₂+···+c _n η _n where η _i are i.i.d. random variables and c _i real constants. We generalize procedure of Haiman (1998) [Haiman, G., 1998, Upper and lower bounds for the tail of the invariant distribution of some AR(1) processes. Asymptotic Methods in Probability and Statistics, 45, 723–730.] to an arbitrary causal linear process and relax the assumptions of his result significantly. This is achieved by investigating the properties of convolution of densities.     

Design selection criteria for discrimination/estimation for nested models and a binomial response

The aim of an experiment is often to enable discrimination between competing forms for a response model. We investigate the selection of a continuous design for a non-sequential strategy when there are two competing generalized linear models for a binomial response, with a common link function and the linear predictor of one model nested within that of the other. A new criterion, _TE$T_E$-optimality, is defined, based on the difference in the deviances from the two models, and comparisons are made with T$T$-, _Ds$D_s$- and D$D$-optimality. Issues are raised through the study of two examples in which designs are assessed using simulation studies of the power to reject the null hypothesis of the smaller model being correct, when the data are generated from the larger model. Parameter estimation for discrimination designs is also discussed and a simple method is investigated of combining designs to form a hybrid design in order to achieve both model discrimination and estimation. This method has a computational advantage over the use of a compound criterion and the similar performance of the designs obtained from the two approaches is illustrated in an example.     

General definition and decomposition of projectors and some applications to statistical problems

A general definition of a set of projectors for decomposing a vector as the sum of vectors belonging to disjoint subspaces not necessarily spanning the whole space is given. Such projectors are defined only over the union of the disjoint subspaces. But their extension to the whole space is of some interest in statistical problems. Explicit expressions are obtained for projectors and their extensions in terms of matrices spanning the subspaces and g-inverses. Decomposition of a projector as the sum of projectors on subspaces is obtained and applied to problems arising in correlation analysis, analysis of variance and estimation of parameters in the Gauss-Markoff model.     

Nonparametric bayesian inference for dichotomous response models

A Bayesian least squares approach is taken here to estimate certain parameters in generalized linear models for dichotomous response data. The method requires that only first and second moments of the probability distribution representing prior information be specified* Examples are presented to illustrate situations having direct estimates as well as those which require approximate or iterative solutions.     

Total least squares solution for compositional data using linear models

The restrictive properties of compositional data, that is multivariate data with positive parts that carry only relative information in their components, call for special care to be taken while performing standard statistical methods, for example, regression analysis. Among the special methods suitable for handling this problem is the total least squares procedure (TLS, orthogonal regression, regression with errors in variables, calibration problem), performed after an appropriate log-ratio transformation. The difficulty or even impossibility of deeper statistical analysis (confidence regions, hypotheses testing) using the standard TLS techniques can be overcome by calibration solution based on linear regression. This approach can be combined with standard statistical inference, for example, confidence and prediction regions and bounds, hypotheses testing, etc., suitable for interpretation of results. Here, we deal with the simplest TLS problem where we assume a linear relationship between two errorless measurements of the same object (substance, quantity). We propose an iterative algorithm for estimating the calibration line and also give confidence ellipses for the location of unknown errorless results of measurement. Moreover, illustrative examples from the fields of geology, geochemistry and medicine are included. It is shown that the iterative algorithm converges to the same values as those obtained using the standard TLS techniques. Fitted lines and confidence regions are presented for both original and transformed compositional data. The paper contains basic principles of linear models and addresses many related problems.     

Some matrix results related to a partitioned singular linear model

Consider the linear model (y, Xβ V), where the model matrix X may not have a full column rank and V might be singular. In this paper we introduce a formula for the difference between the BLUES of Xβ under the full model and the model where one observation has been deleted. We also consider the partitioned linear regression model where the model matrix is (X₁: X₂) the corresponding vector of unknown parameters being (β′₁ : β′₂)′. We show that the BLUE of X₁ β₁ under a specific reduced model equals the corresponding BLUE under the original full model and consider some interesting consequences of this result.     

Log linear models for multivariate paired comparison experiments with ties

A log linear multivariate paired comparison model for ties is proposed in which the cell probabilities under independence are those given by Davidson (1970). Altham's (1970) generalized measure of association (iv) is used to compare the association structure between two models, one having full, the other having reduced association structure. Based on the model with reduced association structure, the analysis of data from a consumer preference experiment is presented.     

Statistical inference for multivariate partially linear regression models

In this paper we study a class of multivariate partially linear regression models. Various estimators for the parametric component and the nonparametric component are constructed and their asymptotic normality established. In particular, we propose an estimator of the contemporaneous correlation among the multiple responses and develop a test for detecting the existence of such contemporaneous correlation without using any nonparametric estimation. The performance of the proposed estimators and test is evaluated through some simulation studies and an analysis of a real data set is used to illustrate the developed methodology. The Canadian Journal of Statistics 41: 1–22; 2013 © 2013 Statistical Society of Canada     

A longitudinal study of children's aggressive behaviours based on multivariate mixed models with incomplete data

The authors consider children's behavioural and emotional problems and their relationships with possible predictors. They propose a multivariate transitional mixed‐effects model for a longitudinal study and simultaneously address non‐ignorable missing data in responses and covariates, measurement errors in covariates, and multivariate modelling of the responses and covariate processes. A real dataset is analysed in details using the proposed method with some interesting results. The Canadian Journal of Statistics 37: 435–452; 2009 © 2009 Statistical Society of Canada     

Robust transformation mixed‐effects models for longitudinal continuous proportional data

The authors propose a robust transformation linear mixed‐effects model for longitudinal continuous proportional data when some of the subjects exhibit outlying trajectories over time. It becomes troublesome when including or excluding such subjects in the data analysis results in different statistical conclusions. To robustify the longitudinal analysis using the mixed‐effects model, they utilize the multivariate t distribution for random effects or/and error terms. Estimation and inference in the proposed model are established and illustrated by a real data example from an ophthalmology study. Simulation studies show a substantial robustness gain by the proposed model in comparison to the mixed‐effects model based on Aitchison's logit‐normal approach. As a result, the data analysis benefits from the robustness of making consistent conclusions in the presence of influential outliers. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Two sample inference in functional linear models

We propose a method of comparing two functional linear models in which explanatory variables are functions (curves) and responses can be either scalars or functions. In such models, the role of parameter vectors (or matrices) is played by integral operators acting on a function space. We test the null hypothesis that these operators are the same in two independent samples. The complexity of the test statistics increases as we move from scalar to functional responses and relax assumptions on the covariance structure of the regressors. They all, however, have an asymptotic chi‐squared distribution with the number of degrees of freedom which depends on a specific setting. The test statistics are readily computable using the R package fda , and have good finite sample properties. The test is applied to egg‐laying curves of Mediterranean flies and to data from terrestrial magnetic observatories. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Consistency of maximum likelihood estimators in a large class of deconvolution models

We consider the maximum likelihood estimator $\hat{F}_n$ of a distribution function in a class of deconvolution models where the known density of the noise variable is of bounded variation. This class of noise densities contains in particular bounded, decreasing densities. The estimator $\hat{F}_n$ is defined, characterized in terms of Fenchel optimality conditions and computed. Under appropriate conditions, various consistency results for $\hat{F}_n$ are derived, including uniform strong consistency. The Canadian Journal of Statistics 41: 98–110; 2013 © 2012 Statistical Society of Canada