Consistency of elementwise-weighted total least squares estimator in a multivariate errors-in-variables model <Emphasis Type="Bold">AX=B</Emphasis>

A multivariate measurement error model AXB is considered. The errors in [A,B] are rowwise independent, but within each row the errors may be correlated. Some of the columns are observed without errors, and in addition the error covariance matrices may differ from row to row. The total covariance structure of the errors is supposed to be known up to a scalar factor. The fully weighted total least squares estimator of X is studied, which in the case of normal errors coincides with the maximum likelihood estimator. We give mild conditions for weak and strong consistency of the estimator, when the number of rows in A increases. The results generalize the conditions of Gallo given for a univariate homoscedastic model (where B is a vector), and extend the conditions of Gleser given for the multivariate homoscedastic model. We derive the objective function for the estimator and propose an iteratively reweighted numerical procedure.Acknowledgements.A. Kukush is supported by a postdoctoral research fellowship of the Belgian office for Scientific, Technical and Cultural Affairs, promoting Scientific and Technical Collaboration with Central and Eastern Europe. S. Van Huffel is a full professor with the Katholieke Universiteit Leuven. This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction (IUAP Phase V-22), initiated by the Belgian State, Prime Ministers Office-Federal Office for Scientific, Technical and Cultural Affairs, of the Concerted Research Action (GOA) projects of the Flemish Government MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology), of the IDO/99/03 project (K.U. Leuven) Predictive computer models for medical classification problems using patient data and expert knowledge, of the FWO projects G.0200.00, G.0078.01 and G.0270.02. The scientific responsibility is assumed by its authors. The authors would like to thank Maria Luisa Rastello and Amedeo Premoli for bringing the EW-TLS problem to their attention. The authors are grateful to two anonymous referees for the valuable comments.     

A characterization of the normal distribution by sufficiency of the least squares estimation

Summary For a linear modelY =ϑ + Z,ϑ ∈V,V ⊂ ℝ ⁿ a linear space, the following theorem is proved under simple conditions on the subspaceV: The projection onV (i.e. the least squares estimate forϑ) is a sufficient statistic iffZ is normally distributed. Further, this result is extended to the case of a multivariate linear model.     

A new interpretation of optimality forE-optimal designs in linear regression models

The optimal design problem for the estimation of several linear combinationsc′ _l ϑ (l=1, …,m) is considered in the usual linear regression modely=f′(x)ϑ (f(x) ∈ ℝ ^k ,ϑ ∈ ℝ ^k ). An optimal design minimizes a (weighted)p-norm of the variances of the least squares estimates for the different linear combinationsc′ _l ϑ. A generalized Elfving theorem is used to derive the relation of the new optimality criterion to theE-optimal design problem. It is shown that theE-optimal design for the parameterϑ minimizes such a (weighted)p-norm whenever the vectorc=(c′ ₁, …, c′_k)′ is an inball vector of a symmetric convex and compact “Elfving set” in.     

An outlier test for linear processes — II. Large contamination

In Flak/Schmid (1993) an outlier test for linear processes was introduced. The test statistic bases on a comparison of each observation with a one-step predictor. It was assumed that an upper bound for the total number of outlierss _n is known, wheren denotes the sample size. The asymptotic distribution of the test statistic was derived under the assumption thats _n/n → 0 ands _n → ∞ asn → ∞. This note deals with the asymptotic behaviour of this quantity, ifs _n/n →p ₀ ∈ (0, 1).     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

A generalized maximum likelihood characterization of the normal distribution

LetP be a probability measure on ℝ andI _x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝⁿ and θ ∈ ℝ the inequality holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization of the normal density to a characterization ofN(0, σ²) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980).     

A Conversation With Paul Embrechts

Paul Embrechts was born in Schoten, Belgium, on 3 February 1953. He holds a Licentiaat in Mathematics from Universiteit Antwerpen (1975) and a DSc from Katholieke Universiteit Leuven (1979), where he was also a Research Assistant from 1975 to 1983. He then held a lectureship in Statistics at Imperial College, London (1983–1985) and was a Docent at Limburgs Universitair Centrum, Belgium (1985–1989) before joining ETH Zürich as a Full Professor of Mathematics in 1989, where he remained until his retirement as an Emeritus in 2018. A renowned specialist of extreme-value theory and quantitative risk management, he authored or coauthored nearly 200 scientific papers and five books, including the highly influential ‘Modelling of Extremal Events for Insurance and Finance’ (Springer, 1997) and ‘Quantitative Risk Management: Concepts, Techniques and Tools’ (Princeton University Press, 2005, 2015). He served in numerous editorial capacities, notably as Editor-in-Chief of the ASTIN Bulletin (1996–2005). Praised for his natural leadership and exceptional communication skills, he helped to bridge the gap between academia and industry through the foundation of RiskLab Switzerland and his sustained leadership for nearly 20 years. He gave numerous prestigious invited and keynote lectures worldwide and served as a member of the board of, or consultant for, various banks, insurance companies and international regulatory authorities. His work was recognised through several visiting positions, including at the Oxford-Man Institute, and many awards. He is, inter alia, an Elected Fellow of the Institute of Mathematical Statistics (1995) and the American Statistical Association (2014), an Honorary Fellow of the Institute and the Faculty of Actuaries (2000), Honorary Member of the Belgian (2010) and French (2015) Institute of Actuaries and was granted four honorary degrees (University of Waterloo, 2007; Heriot-Watt University, 2011; Université catholique de Louvain, 2012; City, University of London, 2017). The following conversation took place in Paul's office at ETH Zürich, 17–18 December 2018.     

A minimal characterization of the covariance matrix

Summary LetX be ak-dimensional random vector with mean vectorμ and non-singular covariance matrix Σ. We show that among all pairs (a, Δ),a ∈ IR ^k , Δ ∈ IR ^k×k positive definite and symmetric andE(X−a)′ Δ⁻¹(X−a)=k, (μ, Σ) is the unique pair which minimizes det Δ. This motivates certain robust estimators of location and scale. Research supported by the Nuffield Foundation.     

Consistent estimation in the presence of incidental parameters

Summary Let (X,A) be a measurable space andP _ϑη |A (ϑη) ∈ Θ x H, ∥A, (θ, η) ∈ Θ×H, a parametrized family of probability measures (for short:p-measures). This paper is concerned with the problem of consistently estimatingθ from realizations governed by , where η^u ∈ H, v ∈ ℕ, are unknown.     

Asymptotic results and tests for the choice of approximative models in nonlinear two-phases regression models, heteroscedatic case

This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t _i)=f(t _i)+ε(t _i), i=1,2,…,n, where the (ε(t _i))_i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time we assume that the ε(t _i) have non constant and unknown variances σ²(t _i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation. Received: February 1996     

A note on multinomial maximum likelihood estimation under ordered restrictions and the EM algorithm

A frequently occurring problem is to find the maximum likelihood estimation (MLE) of p subject to p∈C (C⊂ P the probability vectors in R ^k ). The problem has been discussed by many authors and they mainly focused when p is restricted by linear constraints or log-linear constraints. In this paper, we construct the relationship between the the maximum likelihood estimation of p restricted by p∈C and EM algorithm and demonstrate that the maximum likelihood estimator can be computed through the EM algorithm (Dempster et al. in J R Stat Soc Ser B 39:1–38, 1997). Several examples are analyzed by the proposed method.     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

The Limit Distribution of the Bootstrap for the Unit Root Test Statistic when the Residuals are Dependent

In this article, the unit root test for the AR(1) model with dependent residuals is considered. We adopt a bootstrap procedure to bootstrap the residuals with bootstrap sample size m less than the size n of the original sample. Under the assumptions that m → ∞ and m/n → 0, the convergence in probability of the bootstrap distribution function is established. Research supported by National Natural Science Foundation of China (No. 10471126)     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999     

Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Summary Let the random variableX be normal distributed with known varianceσ ²>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl _p (θ, d)=|θ-d| ^p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.     

Asymptotic expansion of the minimax value in survey sampling

Summary Suppose that a real numbery _u is associated with each unitu of a populationU and that the functiony:u →y _u onU is known to be an element of the parameter space Θ. The statistician has to select a samples ⊂U ofn units and to employy _u;u ∈s to estimate the arithmetic mean of ally _u,u ∈U. The performance of such a strategy is assessed by its mean square error or, more simply, by the supremum of the mean square error. This supremum cannot be determined exactly for the parameter space of Scott/Smith (1975). We propose, therefore, an asymptotic approximation; this approximation is based on the assumption, that the sample sizen is fixed and that linear estimators have to be used.     

Some Decompositions of OLSEs and BLUEs Under a Partitioned Linear Model

We consider in this paper a partitioned linear model { y , X ₁ β ₁ + X ₂ β ₂ , σ ² σ } and two corresponding small models { y , X ₁ β ₁ , σ ² σ } and { y , X ₂ β ₂ , σ ² σ } . We derive necessary and sufficient conditions for (i) the ordinary least squares estimator under the full model to be the sum of the ordinary least squares estimators under the two small models; (ii) the best linear unbiased estimator under the full model to be the sum of the best linear unbiased estimators under the two small models; (iii) the best linear unbiased estimator under the full model to be the sum of the ordinary least squares estimators under the two small models. The proofs of the main results in this paper also demonstrate how to use the matrix rank method for characterizing various equalities of estimators under general linear models.     

An algorithm for panel ANOVA with grouped data

In this paper, we present an algorithm suitable for analysing the variance of panel data when some observations are either given in grouped form or are missed. The analysis is carried out from the perspective of ANOVA panel data models with general errors. The classification intervals of the grouped observations may vary from one to another, thus the missing observations are in fact a particular case of grouping. The proposed Algorithm (1) estimates the parameters of the panel data models; (2) evaluates the covariance matrices of the asymptotic distribution of the time-dependent parameters assuming that the number of time periods, T, is fixed and the number of individuals, N, tends to infinity and similarly, of the individual parameters when T → ∞ and N is fixed; and, finally, (3) uses these asymptotic covariance matrix estimations to analyse the variance of the panel data.     

Conditional nonparametric frontier models for convex and nonconvex technologies: a unifying approach

The explanation of productivity differentials is very important to identify the economic conditions that create inefficiency and to improve managerial performance. In the literature two main approaches have been developed: one-stage approaches and two-stage approaches. Daraio and Simar (2005, J Prod Anal 24(1):93–121) propose a fully nonparametric methodology based on conditional FDH and conditional order-m frontiers without any convexity assumption on the technology. However, convexity has always been assumed in mainstream production theory and general equilibrium. In addition, in many empirical applications, the convexity assumption can be reasonable and sometimes natural. Lead by these considerations, in this paper we propose a unifying approach to introduce external-environmental variables in nonparametric frontier models for convex and nonconvex technologies. Extending earlier contributions by Daraio and Simar (2005, J Prod Anal 24(1):93–121) as well as Cazals et al. (2002, J Econometrics 106:1–25), we introduce a conditional DEA estimator, i.e., an estimator of production frontier of DEA type conditioned to some external-environmental variables which are neither inputs nor outputs under the control of the producer. A robust version of this conditional estimator is proposed too. These various measures of efficiency provide also indicators of convexity which we illustrate using simulated and real data. Cinzia Daraio received Research support from the Italian Ministry of Education Research on Innovation Systems Project (iRis) “The reorganization of the public system of research for the technological transfer: governance, tools and interventions” and from the Italian Ministry of Educational Research Project (MIUR 40% 2004) “System spillovers on the competitiveness of Italian economy: quantitative analysis for sectoral policies” which are acknowledged. Léopold Simar received Research support from the “Interuniversity Attraction Pole”, Phase V (No. P5/24) from the Belgian Government (Belgian Science Policy) is acknowledged.