Criteria for the equality between ordinary least squares and best linear unbiased estimators under certain linear models

A new necessary and sufficient condition is derived for the equality between the ordinary least-squares estimator and the best linear unbiased estimator of the expectation vector in linear models with certain specific design matrices. This condition is then applied to special cases involving one-way and two-way classification models.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

On equalities between BLUES,WLSEs, and SLSEs

Necessary and sufficient conditions for equalities between the best linear unbiased estimator, the weighted least-squares estimator, and the simple least-squares estimator of the expectation vector in a general Gauss-Markoff model are given in some alternative formulations. The main result states, somewhat surprisingly, that the weighted least-squares estimator cannot be identical with the simple least-squares estimator unless they both coincide with the best linear unbiased estimator.     

Let us do the twist again

Krämer (Sankhy $\bar{\mathrm{a }}$ 42:130–131, 1980) posed the following problem: “Which are the $\mathbf{y}$ , given $\mathbf{X}$ and $\mathbf{V}$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $\mathbf{y}$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $\varvec{\beta }$ coincide under the general Gauss–Markov model $\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $\varvec{\beta }$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $\varvec{\beta }$ , we consider the estimation of the systematic part $\mathbf{X} \varvec{\beta }$ , which is a natural consequence of relaxing the assumption that $\mathbf{X}$ and $\mathbf{V}$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $\mathbf{X} \varvec{\beta }$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.     

A note on generalized ridge estimators

It is shown that a necessary and sufficient condition derived by Farebrother (1984)for a generalized ridge estimator to dominate the ordinary least-squares estimator with respect to the mean-square-error-matrix criterion in the linear regression model admits a similar interpretation as the well known criterion of Toro-Viz-carrondo and Wallace (1968)for the dominance of a restricted least-squares estimator over the ordinary least-squares estimator. Two other properties of the generalized ridge estimators, referring to the concept of admissibility, are also pointed out.     

An extension of a rank criterion for the least squares estimator to be the best linear unbiased estimator

Among criteria for the least squares estimator in a linear model (y, Xβ, V) to be simultaneously the best linear unbiased estimator, one convenient for applications is that of Anderson (1971, 1972). His result, however, has been developed under assumptions of full column rank for X and nonsingularity for V. Subsequently, this result has been extended by Styan (1973) to the case when the restriction on X is removed. In this note, it is shown that also the restriction on V can be relaxed and, consequently, that Anderson's criterion is applicable to the general linear model without any rank assumptions at all.     

Nonnegative unbiased estimability of linear combinations of two variance components

For a general mixed model with two variance components θ₁ and θ₂, a criterion for a function q₁θ₁+q₂θ₂ to admit an unbiased nonnegative definite quadratic estimator is established in a form that allows answering the question of existence of such an estimator more explicitly than with the use of the criteria known hitherto. An application of this result to the case of a random one-way model shows that for many unbalanced models the estimability criterion is expressible directly by the largest of the numbers of observations within levels, thus extending the criterion established by LaMotte (1973) for balanced models.     

A complete solution to the problem of robustness of Grubbs's test

Srivastava (1980) showed that Grubbs's test for detecting a univariate outlier is robust against the effect of intraclass correlation structure. Young, Pavur, and Marco (1989) extended this result by proving that both the significance level and the power of Grubbs's test remain unchanged within a wider family of dispersion matrices, introduced by Baldessari (1966) in a different context. In this note, we derive a complete solution of the problem by establishing that the characteristics of Grubbs's test are invariant with respect to a given dispersion matrix if and only if it has Baldessari's structure.     

Sums of squares and products matrices for a non-full ranks hypothesis in the model of potthoff and roy

Formulae for sums of squares and products matrices, useful in testing a general linear hypothesis in the model of POTTHOFF and ROY, are given, Contrary to the customary approach, these formulae are expressed in original terms of the design matrices and the matrices formulating the hypothesis. They are applicable regardless of the ranks of the matrices involved, which allows to avoid a transformation of the hypothesis and a repara-metrization of the model.     

A projector oriented approach to the best linear unbiased estimator

The paper provides a projector based approach to the best linear unbiased estimator (BLUE). By revisiting the so called generalized projection operator, introduced in Rao (J R Stat Soc Ser B Stat Methodol 36:442–448, 1974), a number of new formulae for BLUE is established. Furthermore, some attention is paid to the coincidence of the BLUE and the ordinary least squares estimator.