On the BLUEs in Two Linear Models via C. R. Rao's Pandora's Box

We consider the estimation of the parameters in two partitioned linear models, denoted by 𝒜 = {y, X ₁ β ₁ + X ₂ β ₂, V _𝒜} and ? = {y, X ₁ β ₁ + X ₂ β ₂, V _?}, which we call full models. Correspondingly, we define submodels 𝒜₁ = {y, X ₁ β ₁, V _𝒜} and ?₁ = {y, X ₁ β ₁, V _?}. Using the so-called Pandora's Box approach introduced by Rao (1971 Rao , C. R. ( 1971 ). Unified theory of linear estimation . Sankhy?, Ser. A 33 : 371 – 394 . [Corrigendum (1972), 34, p. 194, 477.]  [Google Scholar], we give new necessary and sufficient conditions for the equality between the best linear unbiased estimators (BLUEs) of X ₁ β ₁ under 𝒜₁ and ?₁ as well as under 𝒜 and ?. In our considerations we will utilise the Frisch–Waugh–Lovell theorem which provides a connection between the full model 𝒜 and the reduced model 𝒜 _r  = {M ₂ y, M ₂ X ₁ β ₁, M ₂ V _𝒜 M ₂} with M ₂ being an appropriate orthogonal projector. Moreover, we consider the equality of the BLUEs under the full models assuming that they are equal under the submodels.     

Missing Values in Linear Regression: Imputations Using An Error-Contaminated Linear Predictor

The problem of missing values problem is common in all branches of statistics and especially in regression analysis. Here we consider estimation of the regression parameters in the presence of missingness in the response. The usual method is to replace the missing value by its predicted value based on the available observations without any correction for the disturbance term. Instead we suggest a method which corrects the usual predictor with a guess of the disturbance term based on the available residuals. Comparison between the two methods shows that the latter leads to better results.     

Elephant-seal movements: Modelling migration

Elephant seals migrate over vast areas of the eastern North Pacific Ocean between rookeries in southern California and distant northern foraging areas. Several models of particle movement were evaluated and a model for great-circle motion found to give reasonable results for the movement of an adult female. This model takes specific account of the fact that the movement is on the surface of a sphere and that the animal is apparently heading toward a particular destination. The parameters of the motion were estimated. Such a great-circle path of migration may imply that these seals have the ability to assess their position with respect to some global or celestial cues, allowing them to continually adjust their course and achieve the most direct geodesic route between origin and destination of migration. But the navigational mechanism actually used by these seals to accomplish such feats is as yet unknown.     

Krishnaiah,P.R. and C.R. Rao (eds):Sampling. Handbook of Statistics,vol. 6

In a recent paper Yager (1988) introduced a special type of aggregation operators which he called OWA-operators and applied it to multicriteria decisionmaking. In our paper we give a more general definition of such operators and investigate in some detail the structural properties of them. In particular we introduce a well motivated ordering relation on weight systems which is different but compatible with Yager's degree of “orness”. Our results have immediate applications to the theory of generalized quantifiers as also discussed by Yager. Details concerning this topic will appear in a future paper.     

Modelling political popularity: a correction

Summary. The paper supplies a missing step in the aggregation argument that was used in our 1997 paper in this journal. We show that monthly poll data representing the average of a sample of voter preferences, evolving in different ways in the face of new information, should follow a fractionally integrated process.     

Estimation in Singular Linear Models with Stochastic Linear Restrictions and Linear Equality Restrictions

This article is concerned with the parameter estimation in a singular linear regression model with stochastic linear restrictions and linear equality restrictions simultaneously. A new estimator is introduced and it is proved that the proposed estimator is superior to the least squares estimator and singular mixed estimator in the mean squared error sense under certain conditions.     

Latent Variable Modelling: A Survey*

Abstract. Latent variable modelling has gradually become an integral part of mainstream statistics and is currently used for a multitude of applications in different subject areas. Examples of ‘traditional’ latent variable models include latent class models, item–response models, common factor models, structural equation models, mixed or random effects models and covariate measurement error models. Although latent variables have widely different interpretations in different settings, the models have a very similar mathematical structure. This has been the impetus for the formulation of general modelling frameworks which accommodate a wide range of models. Recent developments include multilevel structural equation models with both continuous and discrete latent variables, multiprocess models and nonlinear latent variable models.