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• 1.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish Univ Agr Sci, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Univ Rwanda, Rwanda. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data2019Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 48, nr 8, s. 2060-2073Artikel i tidskrift (Refereegranskat)

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. At the first stage, we derive the MSE when the covariance matrices are known. At the second stage, a method based on parametric bootstrap is proposed for bias correction and for prediction error that reflects the uncertainty when the unknown covariance is replaced by its suitable estimator.

• 2.
Stockholm Univ, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish Univ Agr Sci, Sweden.
Small area estimation using reduced rank regression models2019Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415XArtikel i tidskrift (Refereegranskat)

Small area estimation techniques have got a lot of attention during the last decades due to their important applications in survey studies. Mixed linear models and reduced rank regression analysis are jointly used when considering small area estimation. Estimates of parameters are presented as well as prediction of random effects and unobserved area measurements.

• 3.
Department of Statistics, Stockholm University.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences.
Bilinear regression with random effects and reduced rank restrictions2018Rapport (Övrigt vetenskapligt)

Bilinear models with three types of effects are considered: fixed effects, random effects and latent variable effects. Explicit estimators are proposed.

• 4.
Department of Statistics, Stockholm University.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences.
Bilinear regression with rank restrictions on the mean and the dispersion matrix2018Rapport (Övrigt vetenskapligt)

A bilinear regression model with rank restrictions imposed on the mean-parameter matrix and on the dispersion matrix is studied. Maximum likelihood inspired estimates are derived. The approach generalizes classical reduced rank regression analysis and principal component analysis. It is shown via a simulation study and a real example that even for small dimensions the method works as well as reduced rank regression analysis whereas the approach in this article also can be used when the dimension is large.

• 5.
Indian Statistical Institute, Bangalore, India.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Conditional Independence Models which are Totally Ordered2018Rapport (Övrigt vetenskapligt)

The totally ordered conditional independence (TOCI) model N(K) is defined to be the set of all normal distributions on RI such that for each adjacent pair (Ki, Ki+1) $\in$ K, the components of a multivariate normal vector x $\in$ RI, indexed by the set difference { Ki+1 \ Ki } are mutually conditionally independent given the variables indexed by Ki. Here K = {K1 $\subset$ … $\subset$ Kq } is a totally ordered set of subsets of a finite index set I. It is shown that TOCI models constitute a proper subset of lattice conditional independence (LCI) models. It follows that like LCI models, for the TOCI models the likelihood function and parameter space can be factored into the products of conditional likelihood functions and disjoint parameter spaces, respectively, where each conditional likelihood function corresponds to an ordinary multivariate normal regression model.

• 6.
Swedish Natl Food Agcy, Sweden.
Swedish Natl Food Agcy, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish Univ Agr Sci, Sweden. Swedish Natl Food Agcy, Sweden.
Dose-Related Severity Sequence, and Risk-Based Integration, of Chemically Induced Health Effects2018Ingår i: Toxicological Sciences, ISSN 1096-6080, E-ISSN 1096-0929, Vol. 165, nr 1, s. 74-89Artikel i tidskrift (Refereegranskat)

Risk assessment of chemical hazards is typically based on single critical health effects. This work aims to expand the current approach by characterizing the dose-related sequence of the development of multiple (lower- to higher-order) toxicological health effects caused by a chemical. To this end a "reference point profile" is defined as the relation between benchmark doses for considered health effects, and a standardized severity score determined for these effects. For a given dose of a chemical or mixture the probability for exceeding the reference point profile, thereby provoking lower- to higher-order effects, can be assessed. The overall impact at the same dose can also be derived by integrating contributions across all health effects following severity-weighting. In its generalized form the new impact metric relates to the probability of response for the most severe health effects. Reference points (points of departure) corresponding to defined levels of response can also be estimated. The proposed concept, which is evaluated for dioxin-like chemicals, provides an alternative for characterizing the low-dose region below the reference point for a severe effect like cancer. The shape and variability of the reference point profile add new dimensions to risk assessment, which for example extends the characterization of chemical potency, and the concept of acceptable effect sizes for individual health effects. Based on the present data the method shows high stability at low doses/responses, and is also robust to differences in severity categorization of effects. In conclusion, the novel method proposed enables risk-based integration of multiple dose-related health effects. It provides a first step towards a more comprehensive characterization of chemical toxicity, and suggests a potential for improved low-dose risk assessment.

• 7.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. University of Rwanda.
Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data2018Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, s. 1-23Artikel i tidskrift (Refereegranskat)
• 8.
Linnaeus University, Växjö, Sweden.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
On n/p-Asymptotic Distribution of Vector of Weighted Traces of Powers of Wishart Matrices2018Ingår i: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 33, s. 24-40Artikel i tidskrift (Refereegranskat)

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

• 9.
Department of Mathematics, University of Rwanda.
Small area estimation with missing data using a multivariate linear random effects model2018Ingår i: Japanese Journal of Statistics and Data Science, ISSN 2520-8756Artikel i tidskrift (Refereegranskat)

In this article small area estimation with multivariate data that follow a monotonic missing sample pattern is addressed. Random effects growth curve models with covariates are formulated. A likelihood based approach is proposed for estimation of the unknown parameters. Moreover, the prediction of random effects and predicted small area means are also discussed.

• 10.
Poznan University of Life Science, Poland.
Shanghai University of Int Business and Econ, Peoples R China. Stat Sweden, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agriculture Science, Sweden.
Estimation equations for multivariate linear models with Kronecker structured covariance matrices2017Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, nr 16, s. 7902-7915Artikel i tidskrift (Refereegranskat)

The aim of the paper is to determine maximum-likelihood estimation equations. Observations follow a multivariate normal distribution, X-i similar to N-p,N-q (mu, Psi, Sigma), where D[X-i] = Sigma circle times Psi, Psi and Sigma describe the unknown covariance structure between rows and columns of X-i, respectively. Imposing restrictions on Psi and Sigma four types of covariance structures will be considered.

• 11.
High dimensional extension of the growth curve model and its application in genetics2017Ingår i: Statistical Methods & Applications, ISSN 1618-2510, E-ISSN 1613-981X, Vol. 26, nr 2, s. 273-292Artikel i tidskrift (Refereegranskat)

Recent advances in technology have allowed researchers to collect large scale complex biological data, simultaneously, often in matrix format. In genomic studies, for instance, measurements from tens to hundreds of thousands of genes are taken from individuals across several experimental groups. In time course microarray experiments, gene expression is measured at several time points for each individual across the whole genome resulting in a high-dimensional matrix for each gene. In such experiments, researchers are faced with high-dimensional longitudinal data. Unfortunately, traditional methods for longitudinal data are not appropriate for high-dimensional situations. In this paper, we use the growth curve model and introduce test useful for high-dimensional longitudinal data and evaluate its performance using simulations. We also show how our approach can be used to filter genes in time course genomic experiments. We illustrate this using publicly available genomic data, involving experiments comparing normal human lung tissue with vanadium pentoxide treated human lung tissue, designed with the aim of understanding the susceptibility of individuals working in petro-chemical factories to airway re-modelling. Using our method, we were able to filter out 1053 (about 5 %) genes as non-noise genes from a pool of 22,277. Although our focus is on hypothesis testing, we also provided modified maximum likelihood estimator for the mean parameter of the growth curve model and assessed its performance through bias and mean squared error.

• 12.
Stockholm University, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agriculture Science, Sweden. Stockholm University, Sweden.
Influence diagnostics for count data under AB-BA crossover trials2017Ingår i: Statistical Methods in Medical Research, ISSN 0962-2802, E-ISSN 1477-0334, Vol. 26, nr 6, s. 2938-2950Artikel i tidskrift (Refereegranskat)

This paper aims to develop diagnostic measures to assess the influence of data perturbations on estimates in AB-BA crossover studies with a Poisson distributed response. Generalised mixed linear models with normally distributed random effects are utilised. We show that in this special case, the model can be decomposed into two independent sub-models which allow to derive closed-form expressions to evaluate the changes in the maximum likelihood estimates under several perturbation schemes. The performance of the new influence measures is illustrated by simulation studies and the analysis of a real dataset.

• 13.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, College of Science and Technology, University of Rwanda, Kigali, Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Mean-squared errors of small area estimators under a multivariate linear model for repeated measures data2017Rapport (Övrigt vetenskapligt)

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. A two stage estimator of MSE is obtained. At the first stage, we derive the MSE when the covariance matrices are known. To obtain an unbiased estimator of the MSE, at the second stage, a method based on parametric bootstrap is  proposed for bias correction and for prediction error that reects the uncertainty when the unknown covariance is replaced by its suitable estimator.

• 14.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
On E [Pi(k)(i=0) Tr{W-mi}], where W similar to Wp (l, n)2017Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, nr 6, s. 2990-3005Artikel i tidskrift (Refereegranskat)

In this paper, we give a general recursive formula for $\small E[\prod_{i=0}^k Tr\{W^{m_i}\}]$, where $\small W \sim \mathscr{W}_p(I,n)$ denotes a real Wishart matrix. Formulas for fixed n, p  are presented as well as asymptotic versions when $\small \frac{n}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c$i.e. when the so called Kolmogorov condition holds. Finally, we show  application of the asymptotic moment relation when deriving moments for the Marchenko-Pastur distribution (free Poisson law). A numerical  illustration using implementation of the main result is also performed.

• 15.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Dept. Energy and Technology, Swedish Univ. of Agricultural Sci., Uppsala, Sweden.
Dept. Statist.Stockholm Univ., Stockholm, Sweden.
On estimation in some reduced rank extended growth curve models2017Ingår i: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 26, nr 4, s. 299-310Artikel i tidskrift (Refereegranskat)

The general multivariate analysis of variance model has been extensively studied in the statistical literature and successfully applied in many different fields for analyzing longitudinal data. In this article, we consider the extension of this model having two sets of regressors constituting a growth curve portion and a multivariate analysis of variance portion, respectively. Nowadays, the data collected in empirical studies have relatively complex structures though often demanding a parsimonious modeling. This can be achieved for example through imposing rank constraints on the regression coefficient matrices. The reduced rank regression structure also provides a theoretical interpretation in terms of latent variables. We derive likelihood based estimators for the mean parameters and covariance matrix in this type of models. A numerical example is provided to illustrate the obtained results.

• 16.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, College of Science and Technology, University of Rwanda, Kigali, Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Small area estimation under a multivariate linear model for incomplete repeated measures data2017Rapport (Övrigt vetenskapligt)

In this paper, the issue of analysis of multivariate repeated measures data that follow a monotonic sample pattern for small area estimation is addressed. Random effects growth curve models with covariates for both complete and incomplete data are formulated. A conditional likelihood based approach is proposed for estimation of the mean parameters and covariances. Further, the prediction of random effects and predicted small area means are also discussed. The proposed techniques may be useful for small area estimation under longitudinal surveys with grouped response units and drop outs.

• 17.
Department of Mathematics, University of Rwanda. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Small Area Estimation under a Multivariate Linear Model for Repeated measures Data2017Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, nr 21, s. 10835-10850Artikel i tidskrift (Refereegranskat)

In this article, Small Area Estimation under a Multivariate Linear model for repeated measures data is considered. The proposed model aims to get a model which borrows strength both across small areas and over time. The model accounts for repeated surveys, grouped response units and random effects variations. Estimation of model parameters is discussed within a likelihood based approach. Prediction of random effects, small area means across time points and per group units are derived. A parametric bootstrap method is proposed for estimating the mean squared error of the predicted small area means. Results are supported by a simulation study.

• 18.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, College of Science and Technology, University of Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Small area estimation with missing data using a multivariate linear random effects model2017Rapport (Övrigt vetenskapligt)

In this article small area estimation with multivariate data that follow a monotonic missing sample pattern is addressed. Random effects growth curve models with covariates are formulated. A likelihood based approach is proposed for estimation of the unknown  parameters. Moreover, the prediction of random effects and predicted small area means are also discussed.

• 19.
Linnaeus University, Växjö, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Testing Independence via Spectral Moments2017Ingår i: Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009, Vol. 192, s. 263-274Artikel i tidskrift (Refereegranskat)

Assume that a matrix X : p × n is matrix normally distributed and that the Kolmogorov condition, i.e., limn,p→∞ n = c > 0, holds. We propose a test for identity of the covariance matrix using a goodness-of-fit approach. Calculations are based on a recursive formula derived by Pielaszkiewicz et al. The test performs well regarding the power compared to presented alternatives, for both c < 1 or c ≥ 1.

• 20.
Crop yield estimation at district level for agricultural seasons 2014 in Rwanda2016Ingår i: African Journal of Applied Statistics, ISSN 2316-0861, Vol. 3, nr 1, s. 69-90Artikel i tidskrift (Refereegranskat)

In this paper, we discuss an application of Small Area Estimation (SAE) tech- niques under a multivariate linear regression model for repeated measures data to produce district level estimates of crop yield for beans which comprise two varieties, bush beans and climbing beans in Rwanda during agricultural seasons 2014. By using the micro data of National Institute of Statistics of Rwanda (NISR) obtained from the Seasonal Agricul- tural Survey (SAS) 2014 we derive efficient estimates which show considerable gain. The considered model and its estimates may be useful for policy-makers or for further analyses.

• 21.
University of Tartu, Tartu, Estonia.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agricultural Sciences, Uppsala, Sweden. University of Tartu, Tartu, Estonia.
Hypotheses Testing on Covariance Structures: Comparison of likelihood ratio test, Rao's score test and Wald's score test2016Ingår i: Stochastic and Data Analysis Methods and Applications in Statistics and Demography / [ed] James R. Bozeman, Teresa Oliveira and Christos H. Skiadas, 2016, s. 423-425Konferensbidrag (Refereegranskat)

For a normal population likelihood ratio test, Rao’s score test and Wald’s score test for testing covariance structures are compared in the situation when the number of variables and the sample size are growing. Expressions of all three test statistics are derived under the general null-hypothesisΣ=Σ0, using matrix derivative techniques. The special casesΣ=γIpandΣ=Ipare also under consideration. The tests are compared in a simulation experiment with sample sizes varying from 100 to 5000 and dimensionalities from 2 to 50. When the number of variables is growing Rao’s score test behaves most adequately.

• 22.
Swedish University of Agriculture Science, Sweden.
Swedish University of Agriculture Science, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agriculture Science, Sweden.
A two-step estimation method for grouped data with connections to the extended growth curve model and partial least squares regression2015Ingår i: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 139, s. 347-359Artikel i tidskrift (Refereegranskat)

In this article, the two-step method for prediction, which was proposed by Li et al. (2012), is extended for modelling grouped data, which besides having near-collinear explanatory variables, also having different mean structure, i.e. the mean structure of some part of the data is more complex than other parts. In the first step, inspired by partial least squares regression (PLS), the information for explanatory variables is summarized by a multilinear model with Krylov structured design matrices, which for different groups have different size. The multilinear model is similar to the classical growth curve model except that the design matrices are unknown and are functions of the dispersion matrix. Under such a multilinear model, natural estimators for mean and dispersion matrices are proposed. In the second step, the response is predicted through a conditional predictor where the estimators obtained in the first step are utilized. (C) 2015 Elsevier Inc. All rights reserved.

• 23.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, University of Rwanda, Rwanda. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Bilinear regression model with Kronecker and linear structures for the covariance matrix2015Ingår i: Afrika Statistika, ISSN 2316-090X, Vol. 10, nr 2, s. 827-837Artikel i tidskrift (Refereegranskat)

In this paper, the bilinear regression model based on normally distributed random matrix is studied. For these models, the dispersion matrix has the so called Kronecker product structure and they can be used for example to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimating equations in a flip-flop relation are established and the consistency of estimators is studied.

• 24.
Osaka University, Japan.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Swedish University of Agriculture Science, Sweden.
Covariance components selection in high-dimensional growth curve model with random coefficients2015Ingår i: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 136, s. 86-94Artikel i tidskrift (Refereegranskat)

In this paper, the true number of covariance components in a high-dimensional growth curve model with random coefficients are selected. We propose a selection criterion based on a concept from information theory. The proposed criterion satisfies a consistency property of the true covariance components in our high-dimensional setting. The performance of the proposed methodology is illustrated in a simulation study.

• 25.
Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Swed.
Department of Crop Production Ecology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Department of Crop Production Ecology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Swed.
Effects of extreme weather on yield of major arable crops in Sweden2015Rapport (Övrigt vetenskapligt)

Yield data for a series of years on the main crops grown in Sweden were collected and summarised in order to identify years with extremely low yield, determine their frequency and risk level and relate these to weather data in order to identify weather events leading to large yield reductions.

Annual yield data at county level for cereals, field beans, oilseed rape, potatoes and temporary grasses were taken from official statistics for the period 1965-2014. For the period 2005-2012, crop yield data on farm level were also available from official statistics. In addition, yield data for cereals and temporary grasses being studied in long-term experiments (more than 40 years) located in four different agroecological zones of Sweden were considered. Daily temperature and precipitation data for each of the 21 counties in Sweden during the period 1961-2012 were downloaded from the official Swedish weather data website.

In general, yield reductions were higher in northern than in southern counties and higher for spring cereals than winter cereals. Oats, spring rape and potatoes were the crops with the highest yield variation at county level. The frequency of a 30% yield reduction at county level was very low or close to zero in those counties with widespread cereal production, but large reductions occurred in individual years and certain counties (e.g. -80% in Norrbotten county in 1987).

Close agreement between annual area of non-harvested crops and a 30% yield reduction was observed for certain years, crops and counties. The northern counties had on average 4-11% non-harvested crop area, with Norrbotten county having the highest values. The non-harvested area of cereals in southern counties was on average 0-2%.

The risk of severe crop losses on farm level was around 10%, although in a few cases the risk was 25%, depending on the county. More specifically, the overall risk among the counties for individual farms of obtaining 30% lower yield for winter wheat was 5-20%, for spring wheat 5-20%, for rye 5-10% and for spring barley 5-25%. The corresponding risk of obtaining 50% lower yield for oats was 5-20%.

The yield data for individual farms showed large variations, even in years with ‘favourable’ weather conditions. In most years, yield on the lower 10th percentile of farms was less than half the average yield at county level. Winter wheat showed the lowest variation in southern counties and oats and spring rape the highest. Farm-level yield variations were also much higher in Norrbotten county than in southern counties. This large yield variation was confirmed by data from the long-term crop experiments, in which yield reductions exceeding 30% occurred in 5-18% of years (i.e. 2-8 years in the period 1965-2010).

Most years with the lowest yield were associated with a prolonged dry period (<20 mm precipitation over 40 days) and/or a high level of precipitation during the harvesting period (>100 mm during August). However, attempts to correlate county average yields with indices based only on daily temperature and precipitation gave poor and inconsistent results. Similar results were obtained using yield data from the long-term experiments and indices based solely on precipitation.

The large yield variations between individual farms, the heterogeneity of crop responses to Scandinavian weather conditions and the limitations of yield prediction models in terms of detailed input data and result accuracy indicate that yield reductions should be measured on farm level.

Within the study period, precipitation during summer months appeared to increase over time, particularly in 25% of years in southern Sweden. If this situation persists, it will have conflicting effects on crop production, by reducing the risk of drought periods and increasing the risk of rainy harvesting periods.

• 26.
Department of Crop Production Ecology, Swedish University of Agricultural Sciences (SLU), Uppsala, Sweden.
Department of Crop Production Ecology, Swedish University of Agricultural Sciences (SLU), Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Crop Production Ecology, Swedish University of Agricultural Sciences (SLU), Uppsala, Sweden. Department of Crop Production Ecology, Swedish University of Agricultural Sciences (SLU), Uppsala, Sweden.
Effects of extreme weather on yields of major cereal crops in Sweden: Analysis of long-term experiment data2015Ingår i: Aspects of Applied Biology, ISSN 0265-1491, Vol. 128, s. 165-172Artikel i tidskrift (Refereegranskat)

Weather is one of the key factors controlling crop growth and development. To support decision making, it is essential to know how often extreme weather events have affected crop production and the weather indices that cause them. We used long-term experiment data at four locations in Sweden to evaluate the effects of extreme weather on four major cereal crops: winter wheat, spring wheat, barley and oats. Yield reductions during 1965-2010 differed between crops and locations; with greater variation for spring cereals than winter wheat. For about 2-8 years and 1-2 years, out of the 45 years, yield reductions were 30% and 50%, respectively. For these years the total precipitation during early growth and/or harvest time deviated more than 30% from normal more often, than for years with yield reductions less than 30% (or higher yields). However, such deviations in precipitation were common for the whole 46 year period, and using these weather indices as single predictors of yield reductions would fail in the majority of years.

• 27.
Swedish University of Agricultural Sciences, Uppsala, Sweden.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agricultural Sciences, Uppsala, Sweden.
Erratum - A two-step PLS inspired method for linear prediction with group effect.2015Ingår i: Sankhya. Series A: mathematical statistics and probability, Vol. 77, s. 433-436Artikel i tidskrift (Refereegranskat)
• 28.
University of Windsor, Canada. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agriculture Science, Sweden. Stockholm University, Sweden.
Estimation of Several Intraclass Correlation Coefficients2015Ingår i: Communications in statistics. Simulation and computation, ISSN 0361-0918, E-ISSN 1532-4141, Vol. 44, nr 9, s. 2315-2328Artikel i tidskrift (Refereegranskat)

An intraclass correlation coefficient observed in several populations is estimated. The basis is a variance-stabilizing transformation. It is shown that the intraclass correlation coefficient from any elliptical distribution should be transformed in the same way. Four estimators are compared. An estimator where the components in a vector consisting of the transformed intraclass correlation coefficients are estimated separately, an estimator based on a weighted average of these components, a pretest estimator where the equality of the components is tested and then the outcome of the test is used in the estimation procedure, and a James-Stein estimator which shrinks toward the mean.

• 29.
Department of Statistics, Stockholm University.
Explicit estimators in unbalanced mixed linear models 2015Ingår i: Festschrift in honor of professor Ghazi Shukur on the occasion of his 60th birthday / [ed] Thomas Holgersson, Växjö: Linnaeus University Press, Växjö, Sweden , 2015, s. 121-125Kapitel i bok, del av antologi (Övrigt vetenskapligt)
• 30.
Dept. Statist., Stockholm Univ., Stockholm, Sweden; Dept. Automation, Shanghai Jiao Tong Univ., Shanghai, China.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Dept. Energy and Technol., Swedish Univ. Agricultural Sci., Uppsala, Sweden. Dept. Statist.Stockholm Univ., Stockholm, Sweden.
Explicit influence analysis in two-treatment balanced crossover models2015Ingår i: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 24, nr 1, s. 16-36Artikel i tidskrift (Refereegranskat)

This paper considers how to detect influential observations in crossover models with random individual effects. Two influence measures, the delta-beta influence and variance-ratio influence, are utilized as tools to evaluate the influence of the model on the estimates of mean and variance parameters with respect to case-weighted perturbations, which are introduced to the model for studying the ‘influence’ of cases. The paper provides explicit expressions of the delta-beta and variance-ratio influences for the general two-treatment balanced crossover models when the proposed decompositions for the perturbed models hold. The influence measures for each parameter turn out to be closed-form functions of orthogonal projections of specific residuals in the unperturbed model.

• 31.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. University of Rwanda, PO.Box 3900 Kigali, Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE–750 07 Uppsala, Sweden.. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
Extended GMANOVA Model with a Linearly Structured Covariance Matrix2015Rapport (Övrigt vetenskapligt)

In this paper we consider the extended generalized multivariate analysis of variance (GMANOVA) with a linearly structured covariance matrix. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into m + 1 orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

• 32.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. University of Rwanda, Department of Mathematics.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology Swedish University of Agricultural Sciences Uppsala, Sweden.. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Extended GMANOVA Model with a Linearly Structured Covariance Matrix2015Ingår i: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 24, nr 4, s. 280-291Artikel i tidskrift (Refereegranskat)

In this paper we consider the extended generalized multivariate analysis of variance (GMANOVA) with a linearly structured covariance matrix. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into m + 1 orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

• 33.
Institutionen för växtproduktionsekologi, Sveriges lantbruksuniversitet, Uppsala.
Institutionen för skoglig mykologi och växtpatologi, Sveriges lantbruksuniversitet, Uppsala. Institutionen för biomedicin och veterinär folkhälsovetenskap, Sveriges lantbruksuniversitet, Uppsala. Institutionen för växtproduktionsekologi, Sveriges lantbruksuniversitet, Uppsala. Institutionen för kliniska vetenskaper, Sveriges lantbruksuniversitet, Uppsala. Institutionen för Energi och Teknik, Sveriges lantbruksuniversitet, Uppsala. Institutionen för mark och miljö, Sveriges lantbruksuniversitet, Uppsala. Institutionen för husdjurens miljö och hälsa, Sveriges lantbruksuniversitet, Skara. Institutionen för växtbiologi, Sveriges lantbruksuniversitet, Uppsala. Institutionen för mark och miljö, Sveriges lantbruksuniversitet, Uppsala. Institutionen för växtproduktionsekologi, Sveriges lantbruksuniversitet, Uppsala. Institutionen för mark och miljö, Sveriges lantbruksuniversitet, Uppsala. Institutionen för husdjurens utfodring och vård, Sveriges lantbruksuniversitet, Uppsala. Institutionen för biomedicin och veterinär folkhälsovetenskap, Sveriges lantbruksuniversitet, Uppsala. institutionen för Energi och Teknik, Sveriges lantbruksuniversitet, Uppsala. Institutionen för skoglig mykologi och växtpatologi, Sveriges lantbruksuniversitet, Uppsala. Institutionen för kliniska vetenskaper, Sveriges lantbruksuniversitet, Uppsala.
Framtida risker och hot mot svensk spannmåls- respektive mjölkproduktion: En analys av forskningsbehov för att bedöma risker2015Rapport (Övrigt vetenskapligt)

Vad är syftet med denna Riskanalys? Svensk spannmåls- och mjölkproduktion beror på många faktorer av vilka flera är så kallade biofysiska, dvs i allt väsentligt är de av naturvetenskaplig karaktär (t ex väder, sjukdomar mm). En del förändringar i dessa förutsättningar utgör hot. Vår studie avser att identifiera några av dessa hot och utvärdera, utifrån vetenskapligt testade metodiker, sannolikheten för att de orsakar en skada på produktionen. Detta kräver dock ett mycket omfattande arbete och i denna studie har vi därför begränsat oss till att (i) strukturera hur en vetenskapligt baserad riskanalys bör gå till, och (ii) göra ett antal preliminära riskanalyser för att (iii) identifiera kunskapsluckor som behöver forskas på för att analysen ska kunna antas vila på en vetenskaplig grund.

Vad menar vi med Risk? Vi har definierat risk som sannolikheten att ett hot orsakar en viss negativ konsekvens för den skyddsvärda tillgången. Av dessa termer är kanske den sistnämnda den mest centrala. Vad är det vi vill skydda? Vi har valt ut två tillgångar, Sveriges nationella spannmåls- respektive mjölkproduktion och avser då den produktion som lämnar gården, eller används inom gården, och att de skyddas så att de förblir ungefär av den omfattningen de har i dagsläget. Hoten mot denna produktion har valts utifrån förslag från tidigare studier, workshop, tillgången på experter och att hoten ska vara av biofysisk karaktär. Vilket hot som verkligen utgör en stor risk vet vi ju dock inte förrän efter riskanalysen är utförd och valen av hot bygger därför på en preliminär uppskattning. Biofysisk karaktär innebär att vi främst analyserat naturvetenskapliga hot. Hoten orsakar effekter på produktionen i mätbara termer som sedan översätts till en mer abstrakt skala från ingen till extremt negativ konsekvens. Beroende på olika osäkerhetsfaktorer erhåller vi flera konsekvensvärden för ett givet hot, och fördelningen av dessa på konsekvensskalan utgör ett mått på sannolikheten. Risken anges alltså som ett förhållande mellan konsekvens och sannolikhet.

Varför har vi gjort denna systemavgränsning? Riskanalysen har två huvudaktörer; riskhanteraren som definierar vad som ska anlyseras och analysfunktionen som utför analysen. Riskhanteraren är i vårt fall styrgruppen för SLUs forskningsprogram Framtidens lantbruk (FA, 2015) som har definierat typen av hot och de skyddsvärda tillgångarna som ska analyseras. Vi som utfört denna studie är analysfunktionen, och har alltså dessa definitioner som en utgångspunkt. Om vi ändå tillåter oss att spekulera kring valet av spannmåls- respektive mjölkproduktionen så kan det motiveras av SLU's nationella ansvar vad avser den vetenskapliga kompetensen inom de areella näringarna. Ett fokus på biofysiska hot motiveras av att dessa är potentiellt stora och växande, såsom t ex är fallet vad avser klimatförändringar. Riskanalyser av denna typ bildar centrala underlag för att formulera olika strategier, t ex angående livsmedelsförsörjning.

Hur har arbetet gått till? Riskanalyserna har utförts för ett antal "krisscenarier"; fyra avseende hot mot spannmålsproduktionen (Radioaktivt-nedfall, Virus-i-spannmål, Herbicidresistens och Extremt-sommarväder) och tre avseende mjölkproduktionen (Leptospiros-utbrott, Foderimport-stopp och Värmebölja). Analysen tar sin utgångspunkt i ett omvärldsscenario som definierar de yttre förutsättningarna för vad som antas inträffa. Detta ligger till grund för att identifiera troliga hot mot produktionen och vilka åtgärder som förväntas vidtas. Vi har sedan utgått från att dessa hot och åtgärder verkligen har hänt när vi mha våra förklaringsmodeller bestämt effekterna på produktionen i termer av mätbara enheter ("metrics"; t ex procentuell minskning av lokal eller regional veteproduktion). Dessa effekter tolkas/integreras sedan till en konsekvens för, helst den nationella, men i realiteten främst den regionala produktionen i fem nivåer (ingen, liten, måttlig, stor respektive extrem). Osäkerheter i bedömningarna innebär att flera alternativa konsekvenser erhålls, för ett givet hot, och som ligger till grund för en sannolikhetsbedömning. Analyserna har gjorts av experter inom respektive hots vetenskapliga område, men som haft begränsade förutsättningar (av tidsskäl) att göra tillräckligt många bedömningar för att erhålla ett tillförlitligt mått på sannolikhetsfördelningen (osäkerheten). Istället har vi, vilket också är ett huvudsyfte med studien, huvudsakligen försökt identifiera de kunskapsluckor i förklaringsmodellerna som begränsat våra möjligheter att kunna göra vetenskapligt baserade bedömningar av effekterna (se vidare Appendix 3).

Vad är resultatet? Vi har gjort vissa grova skattningar av sannolikheten trots det bristfälliga antalet bedömningar av konsekvenser. Om ett radioaktivt nedfall sker i en region får det extrema konsekvenser för dess spannmålsproduktion på regional nivå. Ett omfattande angrepp av jordburna virus orsakar en måttlig eller stor konsekvens. En utvecklad herbicidresistens hos ogräsen orsakar i huvudsak en liten till måttlig konsekvens. En extremt torr sommar kan ett år orsaka en stor konsekvens och ett annat år ingen alls. Likaledes orsakar en Regnig-sensommar i ca hälften av fallen ingen konsekvens, men för de resterande åren kan alla grader av konsekvenser uppstå på spannmålsproduktionen. För mjölkproduktionen orsakar samtliga tre hot (Leptospiros-utbrott, Foderimport-stopp och Värmebölja) en liten till måttlig konsekvens. Vad avser ett importsopp för foder är detta under förutsättning att olika åtgärdsprogram kombineras. Om fokus läggs på endast ett åtgärdsprogram ökar risken väsentligt. Dessa skattningar ska alltså inte betraktas som en vetenskapligt baserad analys i nuläget, utan demonstrerar främst exempel på resultat från sådana analyser. Skattningarna har hjälpt oss att identifiera vilka kunskaper vi saknar för att analyserna ska kunna betraktas som vetenskapligt baserade (se vidare Tabell 4.3a; sammanfattningar av respektive scenario finns i Resultatdelen).

Analyserna har ibland också lett till att vi identifierat följdhändelser som faller utanför systemavgränsningen för vår studie och som andra studier har till uppgift att utreda. Många av de hot vi analyserat kan leda till betydande ekonomiska konsekvenser för enskilda företag, vilket i sin tur utgör hot mot produktionen. För denna analys krävs dock socioekonomiska analyser. Vi ser här också kopplingar mellan krisscenarier som är av biofysisk karaktär, t ex kan foder kontaminerat med radioaktivt cesium utgöra ett hot mot mjölkproduktionen. Vår studie har dock bara analyserat ett krisscenarios inverkan på antigen spannmåls- eller mjölkproduktionen.

Vilka är de viktigaste slutsatserna? En central fråga är: Hur trovärdiga/säkra är dessa förutsägelser? Risk avser en förutsägelse om något som ännu inte hänt. Det första som behövs är alltså någon form av modell. Dessa modeller kan vara av olika sort i termer av vilken empirisk kunskap de använder för extrapolering (t ex funktioner, behandlingseffekter, mm), om de är objektiva och om de är transparenta. Alla modeller är osäkra i någon mening. Dock saknas i alla de fall vi undersökt mått på modellernas förutsägelseförmåga (med något enstaka undantag). En allmän slutsats blir att forskningen behöver inriktas mot att testa modellernas förutsägelseförmåga mot observationer för att kunna bidra till en vetenskapligt baserad riskanalys av spannmåls- respektive mjölkproduktionen. Detta innebär att experiment- och försöksupplägg behöver göras utifrån hypoteser (modeller) om hur de dynamiska förloppen beror på varierande förutsättningar och omgivningsförhållanden. T ex behöver de statistiska relationerna för hur Extremt-sommarväder påverkar spannmålsproduktionen, som används i vår studie, kompletteras med tester av grödmodeller som kan beakta flera vädervariablers samtida variationer i både tid och rum. För kunskapsluckor som är specifika för respektive hot, se Tabell 4.4.

Sammanfattningsvis behövs (i) fler förutsägelser av respektive potentiellt hots konsekvenser på produktionen, med modeller som har någon form av graderad tillförlitlighet, för att erhålla mått på osäkerheter. Dessutom behövs det (ii) tester av hypoteser för uppskalningar från kontrollerade experiment och försök (på en liten skala i tid och rum) till regional och nationell skala över flera år, och (iii) utveckling av metodiker för hur sannolikheter för hot, åtgärder respektive konsekvenser kan kombineras till en sannolikhetsfördelning som inbegriper bedömningsosäkerheter för alla dessa faktorer. Troligtvis behövs också att fler potentiella biofysiska hot analyseras.

Hur går vi vidare? En mer fullständig riskanalys som inkluderar alla potentiellt stora hot mot produktionen, och samtidigt är vetenskapligt baserad, kräver att potentiella hot utreds kontinuerligt inom respektive produktionsrelaterat ämnesområde vid SLU. Detta kräver troligen att verksamheter som testar hypoteser för att förutsäga effekter av hot knyts nära den experimentella forskningen och experter inom respektive ämnesområde. Det krävs troligen också att en syntesverksamhet etableras på en ämnesövergripande nivå där metodiker kan standardardiseras, och olika hot och dess konsekvenser kan jämföras och kombineras. En sådan fungerande verksamhet behöver utvidga systemgränserna jämfört med vår studie, genom att sannolikheter för att hot uppkommer och att åtgärder faktiskt vidtas, också bedöms. Dessa sannolikheter behöver sedan integreras med sannolikheterna för konsekvenserna på produktionen. Därefter kan riskanalysen utökas till att inbegripa en mer avlägsen framtid, t ex liknande de tidsperspektiv som klimatförändringsanalyser behandlar. Två av hoten mot mjölkproduktionen utgör exempel på riskanalys för en nära framtid (ca 2025). Vi avslutar rapporten med att diskutera hur en sådan ansats kan se ut i ett längre perspektiv.

• 34.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. University of Rwanda, PO.Box 3900 Kigali, Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. bDepartment of Energy and Technology, Swedish University of Agricultural Sciences, SE–750 07 Uppsala, Sweden.. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
Maximum Likelihood Estimation in the Tensor Normal Model with a Structured Mean2015Rapport (Övrigt vetenskapligt)

There is a growing interest in the analysis of multi-way data. In some studies the inference about the dependencies in three-way data is done using the third order tensor normal model, where the focus is on the estimation of the variance-covariance matrix which has a Kronecker product structure. Little attention is paid to the structure of the mean, though, there is a potential to improve the analysis by assuming a structured mean. In this paper, we introduce a 2-fold growth curve model by assuming a trilinear structure for the mean in the tensor normal model and propose an algorithm for estimating parameters. Also, some direct generalizations are presented.

• 35.
Stockholm University, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agriculture Science, Sweden. Stockholm University, Sweden.
On estimation in hierarchical models with block circular covariance structures2015Ingår i: Annals of the Institute of Statistical Mathematics, ISSN 0020-3157, E-ISSN 1572-9052, Vol. 67, nr 4, s. 773-791Artikel i tidskrift (Refereegranskat)

Hierarchical linear models with a block circular covariance structure are considered. Sufficient conditions for obtaining explicit and unique estimators for the variance-covariance components are derived. Different restricted models are discussed and maximum likelihood estimators are presented. The theory is illustrated through covariance matrices of small sizes and a real-life example.

• 36.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE-750 07 Uppsala, Sweden.. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
Recursive formula for E(∏i Tr{(WΣ-1)mi}), where W~Wp(∑; n) in finite and asymptotic regime2015Rapport (Övrigt vetenskapligt)

In this paper, we give a general recursive formula for E(∏i Tr{(WΣ-1)mi}), where W~Wp(∑; n) denotes a real Wishart matrix. Formulas for xed n; p are presented as well as asymptotic versions when n/p→c, when n,p→∞ i.e., when the so called Kolmogorov condition holds. Finally, we show application of the asymptotic moment relation when deriving moments for the Marchenko-Pastur distribution (free Poisson law). A numerical illustration using implementation of the main result is also performed.

• 37.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Department of Mathematics, College of Science and Technology, University of Rwanda, P.O. Box 3900 Kigali, Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Department of Mathematics, College of Science and Technology, University of Rwanda, P.O. Box 3900 Kigali, Rwanda. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE- 750 07 Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
Small Area Estimation under a Multivariate Linear Model for Repeated Measures Data2015Rapport (Övrigt vetenskapligt)

In this paper, we consider small area estimation under a multivariate linear regression model for repeated measures data. The aim of the proposed model is to get a model which borrows strength across small areas and over time, by incorporating simultaneously the area effects and time correlation. The model accounts for repeated surveys, group individuals and random effects variations. Estimation of model parameters is discussed within a restricted maximum likelihood based approach. Prediction of random e ects and the prediction of small area means across time points and per group units for all time points are derived. The results are supported by a simulation study.

• 38.
Uppsala University, Sweden; Swedish University of Agriculture Science, Sweden; University of Munich, Germany.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Swedish University of Agriculture Science, Sweden.
Tests for high-dimensional covariance matrices using the theory of U-statistics2015Ingår i: Journal of Statistical Computation and Simulation, ISSN 0094-9655, E-ISSN 1563-5163, Vol. 85, nr 13, s. 2619-2631Artikel i tidskrift (Refereegranskat)

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p and#8811;n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.

• 39.
Uppsala University, Sweden; Swedish University of Agriculture Science, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Swedish University of Agriculture Science, Sweden.
Tests of Covariance Matrices for High Dimensional Multivariate Data Under Non Normality2015Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 44, nr 7, s. 1387-1398Artikel i tidskrift (Refereegranskat)

Ahmad et al. (in press) presented test statistics for sphericity and identity of the covariance matrix of a multivariate normal distribution when the dimension, p, exceeds the sample size, n. In this note, we show that their statistics are robust to normality assumption, when normality is replaced with certain mild assumptions on the traces of the covariance matrix. Under such assumptions, the test statistics are shown to follow the same asymptotic normal distribution as under normality for large p, also whenp greater thangreater than n. The asymptotic normality is proved using the theory of U-statistics, and is based on very general conditions, particularly avoiding any relationship between n and p.

• 40.
Universidad Nacional Autonoma de Mexico (UNAM), Mexico.
Universidad Nacional Autonoma de Mexico (UNAM), Mexico. Swedish University of Agricultural Sciences, Uppsala, Sweden.
A new method to compare statistical tree growth curves: The PL-GMANOVA model and its application with dendrochronological data2014Ingår i: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 9, artikel-id e112396Artikel i tidskrift (Refereegranskat)

Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q= A?T+E, where for b=0 : Q~Ei½{b: r{Ei½{b: r1 and for b~0 : Q~Ln½r=r1, A = initial relative growth to be estimated, T~t{t1, and E is an error term for each tree and time point. Furthermore, Ei[–b?r] = Ð (Exp½{b: r=r)dr, b~{1=TPR, with TPR being the turning point radius in a sigmoid curve, and r1 at t1 is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth A^. One site (at the Popocate´petl volcano) stood out, with A^ being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocate´petl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time.

• 41.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
Cumulant-moment relation in free probability theory2014Ingår i: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 18, nr 2, s. 265-278Artikel i tidskrift (Refereegranskat)

The goal of this paper is to present and prove a cumulant-moment recurrent relation formula in free probability theory. It is convenient tool to determine underlying compactly supported distribution function. The existing recurrent relations between these objects require the combinatorial understanding of the idea of non-crossing partitions, which has been considered by Speicher and Nica. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts. Similarly like the non-recursive formulation of Mottelson our formula demands only summing over partitions of the set. The proof of non-recurrent result is given with use of Lagrange inversion formula, while in our proof the calculations of the Stieltjes transform of the underlying measure are essential.

• 42.
Stockholm University, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Stockholm University, Sweden.
Local Influence Analysis in AB-BA Crossover Designs2014Ingår i: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 41, nr 4, s. 1153-1166Artikel i tidskrift (Refereegranskat)

The aim of this article is to develop methodology for detecting influential observations in crossover models with random individual effects. Various case-weighted perturbations are performed. We obtain the influence of the perturbations on each parameter estimator and on their dispersion matrices. The obtained results exhibit the possibility to obtain closed-form expressions of the influence using the residuals in mixed linear models. Some graphical tools are also presented.

• 43.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE-750 07 Uppsala, Sweden.. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan.
On Free Moments and Free Cumulants2014Rapport (Övrigt vetenskapligt)

The concepts of free cumulants and free moments are indispensably related to the idea of freeness introduced by Voiculescu [Voiculescu, D., Proc. Conf., Buşteni/Rom., Lect. Notes Math. 1132(1985), pp. 556-588] and studied further within Free probability theory. Free probability theory is of great importance for both the developing mathematical theories as well as for problem solving methods in engineering.

The goal of this paper is to present theoretical framework for free cumulants and moments, and then prove a new free cumulant-moment relation formula. The existing relations between these objects will be given. We consider as drawback that they require the combinatorial understanding of the idea of non--crossing partitions, which has been considered by Speicher [Speicher, R., Math. Ann., 298(1994), pp. 611-628] and then widely studied and developed by Speicher and Nica [Nica, A. and Speicher, R.:  Lectures on the Combinatorics of Free Probability, Cambridge University Press, Cambridge, United Kingdom, 2006]. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts, instead the calculations of the Stieltjes transform of the underlying measure are essential.

The presented free cumulant--moment relation formula is used to calculate cumulants of degree 1 to 5 as a function of the moments of lower degrees. The simplicity of the calculations can be observed by a comparison with the calculations performed in the classical way using non-crossing partitions. Then, the particular example of non-commutative space i.e., space of p×p matrices X=(Xij)ij, where Xij has finite moments, equipped with functional E(TrX)∕p is investigated.

• 44.
Swedish University of Agricultural Sciences, Uppsala, Sweden and Department of Statistics, Uppsala University, Sweden.
A note on mean testing for high dimensional multivariate data under non-normality2013Ingår i: Statistica neerlandica (Print), ISSN 0039-0402, E-ISSN 1467-9574, Vol. 67, nr 1, s. 81-99Artikel i tidskrift (Refereegranskat)

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n,p→∞, and under mild assumptions, but without assuming any relationship between n and p, the statistic is shown to asymptotically follow a chi-square distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of the well-known Box's approximation, under high-dimensional set up. Using a classical limit theorem, the approximation is further extended to an asymptotic normal limit under the same high dimensional set up. The simulation results, generated under different parameter settings, are used to show the accuracy of the approximation for moderate n and large p.

• 45.
Swedish University of Agricultural Sciences, Uppsala, Sweden.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
A two-step PLS inspired method for linear prediction with group effect.2013Ingår i: Sankhya Ser A, ISSN 0976-836X, Vol. 75, s. 96-117Artikel i tidskrift (Refereegranskat)

In this article, we consider prediction of a univariate response from background data. The data may have a near-collinear structure and additionally group effects are assumed to exist. A two-step method is proposed. The first step summarizes the information in the predictors via a bilinear model. The bilinear model has a Krylov structured within individual design matrix, which is the link to classical partial least squares (PLS) analysis and a between-individual design matrix which handles group effects. The second step is the prediction step where a conditional expectation approach is used. The two-step method gives new insight into PLS. Explicit maximum likelihood estimators of the dispersion matrix and mean for the predictors are derived under the assumption that the covariance between the response and explanatory variables is known. It is shown that for within-sample prediction the mean squared error of the two-step method is always smaller than PLS

• 46.
Department of Statistics, Stockholm University, SE–106 91 Stockholm, Sweden.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Department of Statistics, Stockholm University, SE–106 91 Stockholm, Sweden.
Hierarchical Models with Block Circular Covariance Structures2013Rapport (Övrigt vetenskapligt)

Hierarchical linear models with a block circular covariance structure are considered. Sufficient conditions for obtaining explicit and unique estimators for the variance-covariance components are derived. Different restricted models are discussed and maximum likelihood estimators are presented.

• 47.
The Multilinear Normal Distribution: Introduction and Some Basic Properties2013Ingår i: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 113, nr S1, s. 37-47Artikel i tidskrift (Refereegranskat)

In this paper, the multilinear normal distribution is introduced as an extension of the matrix-variate normal distribution. Basic properties such as marginal and conditional distributions, moments, and the characteristic function, are also presented.

The estimation of parameters using a flip-flop algorithm is also briefly discussed.

• 48.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska högskolan. Swedish University of Agricultural Sciences.
Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix2012Ingår i: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 16, nr 1, s. 13-32Artikel i tidskrift (Refereegranskat)

In this paper the extended growth curve model with two terms and a linearly structured covariance matrix is considered. We propose an estimation procedure that handles linear structured covariance matrices. The idea is first to estimate the covariance matrix when it should be used to define an inner product in a regression space and thereafter reestimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into three orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed and numerical examples are given.

• 49.
Swedish University of Agriculture Science, Sweden .
Maximum Likelihood Estimators in a Two Step Model for PLS2012Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, nr 13-14, s. 2503-2511Artikel i tidskrift (Refereegranskat)

Univariate partial least squares regression (PLS1) is a method of modeling relationships between a response variable and explanatory variables, especially when the explanatory variables are almost collinear. The purpose is to predict a future response observation, although in many applications there is an interest to understand the contributions of each explanatory variable. It is an algorithmic approach. In this article, we are going to use the algorithm presented by Helland (1988). The population PLS predictor is linked to a linear model including a Krylov design matrix and a two-step estimation procedure. For the first step, the maximum likelihood approach is applied to a specific multivariate linear model, generating tools for evaluating the information in the explanatory variables. It is shown that explicit maximum likelihood estimators of the dispersion matrix can be obtained where the dispersion matrix, besides representing the variation in the error, also includes the Krylov structured design matrix describing the mean.

• 50.
More on the Kronecker Structured Covariance Matrix2012Ingår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, nr 13-14, s. 2512-2523Artikel i tidskrift (Refereegranskat)

In this paper, the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly focused is the estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The suggested estimation generalizes the procedure proposed by Srivastava et al. (2008) for a separable covariance matrix. The restrictions imposed by separability and double separability are also discussed.

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