In this paper least squares and maximum likelihood estimates under univariate and multivariate linear models with a priori information related to maximum effects in the models are determined. Both loss functions (the least squares and negative log-likelihood) and the constraints are convex, so the convex optimization theory can be utilized to obtain estimates, which in this paper are called Safety belt estimates. In particular, the complementary slackness condition, common in convex optimization, implies two alternative types of solutions, strongly dependent on the data and the restriction.

It is experimentally shown that, despite of the similarity to the ridge regression estimation under the univariate linear model, the Safety belt estimates behave usually better than estimates obtained via ridge regression. Moreover, concerning the multivariate model, the proposed technique represents a completely novel approach.

The likelihood approach used in this paper leads to quadratic discriminant functions. Classification into one of two known multivariate normal populations with a known and unknown covariance matrix are separately considered, where the two cases depend on the sample size and an unknown squared Mahalanobis distance. Their exact distributions are complicated to obtain. Therefore, moments for the likelihood based discriminant functions are established to express the basic characteristics of respective distribution.

The late Professor Heinz Neudecker (1933-2017) made significant contributions to the development of matrix differential calculus and its applications to econometrics, psychometrics, statistics, and other areas. In this paper, we present an insightful overview of matrix-oriented findings and their consequential implications in statistics, drawn from a careful selection of works either authored by Professor Neudecker himself or closely aligned with his scientific pursuits. The topics covered include matrix derivatives, vectorisation operators, special matrices, matrix products, inequalities, generalised inverses, moments and asymptotics, and efficiency comparisons within the realm of multivariate linear modelling. Based on the contributions of Professor Neudecker, several results related to matrix derivatives, statistical moments and the multivariate linear model, which can literally be considered to be his top three areas of research enthusiasm, are particularly included.

In this paper, we present a statistical approach to evaluate the relationship between variables observed in a two-factors experiment. We consider a three-level model with covariance structure Sigma circle times psi(1) circle times psi(2), where Sigma is an arbitrary positive definite covariance matrix, and psi(1) and psi(2) are both correlation matrices with a compound symmetric structure corresponding to two different factors. The Rao's score test is used to test the hypotheses that observations grouped by one or two factors are uncorrelated. We analyze a fermentation process to illustrate the results. Supplementary materials accompanying this paper appear online.

In this paper, probabilities of misclassification of a two-step likelihood-based discriminant rule are established for the classification of growth curves. The defined two-step classifier considers the fact that the growth curves might not belong to any of the two predetermined populations. The distribution for the classifier is approximated via an Edgeworth-type expansion.

The possibility that a new observation can be allocated to an unknown population is considered. von Rosen and Singull (2022) derived a classi cation rule taking into account this perspective. The classi cation rule consists of two criteria. In this paper, the mean and variance of these criteria needed to discriminate between two growth curves are established.

In this article, the GMANOVA-MANOVA model is considered. Two different matrix residuals are established. The interpretation of the residuals is discussed and several properties are verified. A data set illustrates how the residuals can be used.

The exact distribution of a classification function is often complicated to allow for easy numerical calculations of misclassification errors. The use of expansions is one way of dealing with this diculty. In this paper, approximate probabilities of misclassification of the maximum likelihood based discriminant function are established via an Edgeworth-type expansion based on the standard normal distribution for discriminating between two multivariate normal populations.

In this paper we consider discrimination between two populations of repeated measurements using growth curve models. We establish a classification procedure that is likelihood based, in that sense that we compare the two likelihoods given that the new observation belongs to respectively population. We also discuss the possibility that we classify the new observation to an unknown population, which we show is natural when considering growth curves.

Panel survey data from Uganda, as well as data from the 2014 Uganda Population and Housing Census data have been analysed. The Growth Curve model with rank restrictions on parameters was used to estimate the small area means.

The aim of the analysis was to assess change over time in household living standards (welfare), i.e., to investigate whether households display growth in living standards? whether households grow at the same rates? and whether households in different geographical areas of the country grow at the same rates?

Using a GMANOVA-MANOVA model with rank restrictions on parameters, it was established that growth in household standards of living in Uganda varied across small areas. Sub-regions (small areas) with the highest standards of living in Uganda at the endline were Central Urban region, Kampala Urban region and South Western Urban region, while the sub-regions with the lowest standards of living at the endline were North East Rural region, North East Urban region and Eastern Rural (region). The sub-regions with the highest growth rates in standards of living were Mid West Urban region, Mid North Rural region, and South Western Urban region. The sub-regions with the highest decline in standards of living were East Central Rural region, East Rural region and West Nile Urban region.