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• 1.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A U-statistics Based Approach to Mean Testing for High Dimensional Multivariate Data Under Non-normality2011Report (Other academic)

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 large, and under mild assumptions, the statistic is shown to asymptotically follow a normal distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of well-known Box's approximation, under high-dimensional set up.

• 2.
Uppsala University, Sweden; Swedish University of Agriculture Science, Sweden; University of Munich, Germany.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agriculture Science, Sweden.
Tests for high-dimensional covariance matrices using the theory of U-statistics2015In: Journal of Statistical Computation and Simulation, ISSN 0094-9655, E-ISSN 1563-5163, Vol. 85, no 13, p. 2619-2631Article in journal (Refereed)

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.

• 3.
Uppsala University, Sweden; Swedish University of Agriculture Science, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Swedish University of Agriculture Science, Sweden.
Tests of Covariance Matrices for High Dimensional Multivariate Data Under Non Normality2015In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 44, no 7, p. 1387-1398Article in journal (Refereed)

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.

• 4.
Swedish University of Agricultural Sciences, Uppsala, Sweden and Department of Statistics, Uppsala University, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A note on mean testing for high dimensional multivariate data under non-normality2013In: Statistica neerlandica (Print), ISSN 0039-0402, E-ISSN 1467-9574, Vol. 67, no 1, p. 81-99Article in journal (Refereed)

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.

• 5.
Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Asymptotic distribution of the estimators of a harmonic component in a multivariate time series under m-dependence2011Report (Other academic)

Multivariate time series with definite harmonic structure is considered, in the special case when the marginal univariate time series are independent and asymptotically stationary to second order. The asymptotic distribution of the estimators of a harmonic component under $m$-dependence is found

• 6.
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 Sweden2015Report (Other academic)

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.

• 7.
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 risker2015Report (Other academic)

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.

• 8.
University of Windsor, Canada. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agriculture Science, Sweden. Stockholm University, Sweden.
Estimation of Several Intraclass Correlation Coefficients2015In: Communications in statistics. Simulation and computation, ISSN 0361-0918, E-ISSN 1532-4141, Vol. 44, no 9, p. 2315-2328Article in journal (Refereed)

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.

• 9.
Indian Statistical Institute, Bangalore, India.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Conditional Independence Models which are Totally Ordered2018Report (Other academic)

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.

• 10.
Dept. Statist., Stockholm Univ., Stockholm, Sweden; Dept. Automation, Shanghai Jiao Tong Univ., Shanghai, China.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Dept. Energy and Technol., Swedish Univ. Agricultural Sci., Uppsala, Sweden. Dept. Statist.Stockholm Univ., Stockholm, Sweden.
Explicit influence analysis in two-treatment balanced crossover models2015In: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 24, no 1, p. 16-36Article in journal (Refereed)

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.

• 11.
Stockholm University, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Stockholm University, Sweden.
Local Influence Analysis in AB-BA Crossover Designs2014In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 41, no 4, p. 1153-1166Article in journal (Refereed)

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.

• 12.
Osaka University, Japan.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Swedish University of Agriculture Science, Sweden.
Covariance components selection in high-dimensional growth curve model with random coefficients2015In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 136, p. 86-94Article in journal (Refereed)

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.

• 13.
Graduate School of Science, Hiroshima University, Japan.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences, Sweden.
On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix2019Report (Other academic)

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. We complement the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart  matrix are close to the degrees of freedom of the distribution.

• 14.
University of Tartu, Tartu, Estonia.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. 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 test2016In: Stochastic and Data Analysis Methods and Applications in Statistics and Demography / [ed] James R. Bozeman, Teresa Oliveira and Christos H. Skiadas, 2016, p. 423-425Conference paper (Refereed)

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.

• 15.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Ohlson, MartinLinköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.von Rosen, DietrichLinköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Proceedings of Workshops on Inverse Problems, Data, Mathematical Statistics and Ecology2011Conference proceedings (editor) (Other academic)

Processes in Nature may be considered as deterministic or/and random. We are observing global problems such as climate changes (e.g. warming and extreme weather conditions), pollutions (e.g. acidification, fertilization, the spread of many types of pollutants through air and water) and whole ecosystems that are under pressure (e.g. the Baltic sea and the Arctic region). To understand the processes in Nature and (predict) understand what might occur it is not enough with empirical studies. One needs theoretical fundaments including models and theories to perform correct actions against different threats or at least to carry out appropriate simulation studies. For example, extreme value theory can explain some of the observed phenomena, classical risk analysis may be of help, different types of multivariate and high-dimensional analysis can explain data, time series analysis is essential, for forthcoming studies the theory of experimental designs is of interest, data assimilation together with inverse problem technique is useful for adjustment of data into mathematical models and the list can be made much longer. Behind all these approaches mathematics is hidden, sometimes at a very advanced level. Chemical and physical processes influence all observations but the challenge is to do appropriate approximations so that mathematical/statistical models can be applied. The main aim of this project is to present state of the art knowledge concerning the modelling of Nature with focus on mathematical modelling, in particular "inverse and ill-posed problems", as well as spatiotemporal models. Inverse and ill-posed problems are characterized by the property that the solutions are extremely sensitive to measurement and modelling errors. There are established connections between inverse problems and Bayesian inference but very little has been carried out with focus on parametric inference such as the likelihood approach. Concerning spatio-temporal models these are usually extensions of classical time series models or/and classical multivariate analysis models.

From the Nordic Council of Ministers, within the program Nordic - Russian Cooperation in Education and Research we asked for funding of 3 preparatory meetings where the plan was to create a series of events taking place during 2011-2013. Partner organizations were

• Institute of Problems of Mechanical Engineering, St. Petersburg
• St. Petersburg State University
• Helsinki University
• Swedish Agricultural University
• Stockholm University

However, there were also some other participants from other universities.

The planned events should be connected to the following fields: applied mathematics, biophysics and mathematical statistics. Within applied mathematics: mathematical modelling and partial differential equations, inverse and ill-pose problems, data assimilation, dynamical systems, linear algebra, matrix theory; within biophysics; neural networks and inverse modelling of objects; within mathematical statistical; analyses of stochastic processes, spatio-temporal modelling, experimental design, where considered. There exists a wide overlap between these areas and it is challenging to systemize this overlap and transmit this knowledge to students and stakeholders. However, due to unsure funding it was decided to discuss what can be presented during a one-year program. Moreover, due to practical reasons only 2 meetings/workshops were held:

1. Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology: May 20-21, 2010 at Department of Mathematics, Linköping University.
2. Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology, Part II: August 25-27, 2010 at Department of Mathematics, Helsinki University.

The output from the above events can be summarized as follows:

• We have identified a number of different areas which can be taught on from different perspectives depending on students background of mathematics.
• We have learned to know many interesting researchers who are willing to share there experiences when for example creating a summer school.
• There is no doubt that we can organize cross-disciplinary summer/winter schools with focus on either the Baltic or Archtic regions.

This booklet is also part of the deliverables. It comprizes extended abstracts of the majority of the talks of the participants showing their great interest. It is in some way a unique cross-disciplinary document which has joined researchers from different areas from Russia, Finland and Sweden.

We are extremely grateful for the support given by the Nordic Council of Ministers (NCM-RU-PA-2009/10382) and all the enthusiastic contributions by the participants, including our host in Helsinki, professor Lassi Päivärinta.

• 16.
Swedish University of Agriculture Science, Sweden.
Swedish University of Agriculture Science, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. 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 regression2015In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 139, p. 347-359Article in journal (Refereed)

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.

• 17.
Swedish University of Agricultural Sciences, Uppsala, Sweden.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
A two-step PLS inspired method for linear prediction with group effect.2013In: Sankhya Ser A, ISSN 0976-836X, Vol. 75, p. 96-117Article in journal (Refereed)

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

• 18.
Swedish University of Agricultural Sciences, Uppsala, Sweden.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agricultural Sciences, Uppsala, Sweden.
Erratum - A two-step PLS inspired method for linear prediction with group effect.2015In: Sankhya. Series A: mathematical statistics and probability, Vol. 77, p. 433-436Article in journal (Refereed)
• 19.
Energy and Technology, Swedish University of Agricultural Sciences, Uppsala.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A Two Step Model for Linear Prediction with Group Effect2011Report (Other academic)

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 estimation procedure is proposed. The first step is to summarize 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 approach gives new insight in PLS. Explicit maximum likelihood estimator of the dispersion matrix and mean for the predictors are derived under the assumption that the covariance between the response and explanatory variable is known. It is shown that the mean square error of the two step approach is always smaller than PLS.

• 20.
Swedish University of Agriculture Science, Sweden .
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Maximum Likelihood Estimators in a Two Step Model for PLS2012In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, no 13-14, p. 2503-2511Article in journal (Refereed)

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.

• 21.
Department of Statistics, Stockholm University, SE–106 91 Stockholm, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Statistics, Stockholm University, SE–106 91 Stockholm, Sweden.
Hierarchical Models with Block Circular Covariance Structures2013Report (Other academic)

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.

• 22.
Stockholm University, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agriculture Science, Sweden. Stockholm University, Sweden.
On estimation in hierarchical models with block circular covariance structures2015In: Annals of the Institute of Statistical Mathematics, ISSN 0020-3157, E-ISSN 1572-9052, Vol. 67, no 4, p. 773-791Article in journal (Refereed)

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.

• 23.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Mathematics, College of Science and Technology, University of Rwanda, P.O. Box 3900 Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Mathematics, College of Science and Technology, University of Rwanda, P.O. Box 3900 Kigali, Rwanda. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE- 750 07 Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Small Area Estimation under a Multivariate Linear Model for Repeated Measures Data2015Report (Other academic)

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.

• 24.
Department of Mathematics, University of Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Small area estimation with missing data using a multivariate linear random effects model2018In: Japanese Journal of Statistics and Data Science, ISSN 2520-8756Article in journal (Refereed)

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.

• 25.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, College of Science and Technology, University of Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Small area estimation with missing data using a multivariate linear random effects model2017Report (Other academic)

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.

• 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. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. 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 data2015In: Aspects of Applied Biology, ISSN 0265-1491, Vol. 128, p. 165-172Article in journal (Refereed)

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.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Swedish University of Agricultural Sciences.
Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix2012In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 16, no 1, p. 13-32Article in journal (Refereed)

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.

• 28.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, University of Rwanda, Rwanda. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Bilinear regression model with Kronecker and linear structures for the covariance matrix2015In: Afrika Statistika, ISSN 2316-090X, Vol. 10, no 2, p. 827-837Article in journal (Refereed)

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.

• 29.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Swedish University of Agricultural Sciences. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Estimation in multivariate linear models with Kronecker product and linear structures on the covariance matricesManuscript (preprint) (Other academic)

This paper deals with models based on normally distributed random matrices. More specifically the model considered is X ∼ Np,q(M, Σ, Ψ) with mean M, a p×q matrix, assumed to follow a bilinear structure, i.e., E[X] = M = ABC, where A and C are known design matrices, B is unkown parameter matrix, and the dispersion matrix of X has a Kronecker product structure, i.e., D[X] = Ψ ⊗ Σ, where both Ψ and Σ are unknown positive definite matrices. The model may be used for example to model data with spatiotemporal relationships. The aim is to estimate the parameters of the model when, in addition, Σ is assumed to be linearly structured. In the paper, on the basis of n independent observations on the random matrix X, estimation equations in a flip-flop relation are presented and numerical examples are given.

• 30.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. University of Rwanda, PO.Box 3900 Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE–750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Extended GMANOVA Model with a Linearly Structured Covariance Matrix2015Report (Other academic)

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.

• 31.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. University of Rwanda, Department of Mathematics.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology Swedish University of Agricultural Sciences Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Extended GMANOVA Model with a Linearly Structured Covariance Matrix2015In: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 24, no 4, p. 280-291Article in journal (Refereed)

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.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. University of Rwanda, PO.Box 3900 Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. bDepartment of Energy and Technology, Swedish University of Agricultural Sciences, SE–750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Maximum Likelihood Estimation in the Tensor Normal Model with a Structured Mean2015Report (Other academic)

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.

• 33.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
More on the Kronecker Structured Covariance Matrix2012In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, no 13-14, p. 2512-2523Article in journal (Refereed)

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.

• 34.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics .
More on the Kronecker Structured Covariance Matrix2011Report (Other academic)

In this paper the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly, estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The estimation procedure, suggested in this paper, is a generalization of the procedure derived by Srivastava et al. (2008), for a separable covariance matrix.

Furthermore, the restrictions imposed by separability and double separability are discussed.

• 35.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
The Multilinear Normal Distribution: Introduction and Some Basic Properties2013In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 113, no S1, p. 37-47Article in journal (Refereed)

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.

• 36.
Linköping University, Department of Mathematics. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
The Multilinear Normal Distribution:Introduction and Some Basic Properties2011Report (Other academic)

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 briefy discussed.

• 37.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Explicit Estimators of Parameters in the Growth Curve Model with Linearly Structured Covariance Matrices2009Report (Other academic)
• 38.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Explicit Estimators of Parameters in the Growth Curve Model with Linearly Structured Covariance Matrices2009Conference paper (Other academic)

Estimation of parameters in the classical Growth Curve model when the covariance matrix has some specific linear structure is considered. In our examples maximum likelihood estimators can not be obtained explicitly and must rely on optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linear structured covariance matrix.

• 39.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
More on the Kronecker structured covariance matrix2010Conference paper (Other academic)

The Kronecker structured covariance matrix in multivariate normal distribution will be studied. Particularly, the mapping and parametrization which are induced by the Kronecker product are considered.

Furthermore, estiamtion and the uniqueness of the estimators will be discussed in the case of a covariance matrix which is a Kronecker product of several matrices.

• 40.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Cumulant-moment relation in free probability theory2014In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 18, no 2, p. 265-278Article in journal (Refereed)

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.

• 41.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
On E [Pi(k)(i=0) Tr{W-mi}], where W similar to Wp (l, n)2017In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, no 6, p. 2990-3005Article in journal (Refereed)

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.

• 42.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE-750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
On Free Moments and Free Cumulants2014Report (Other academic)

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.

• 43.
Linnaeus University, Växjö, Sweden.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
On n/p-Asymptotic Distribution of Vector of Weighted Traces of Powers of Wishart Matrices2018In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 33, p. 24-40Article in journal (Refereed)

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 $p$.

• 44.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
On p/n-asymptoticsapplied to traces of 1st and 2nd order powers of Wishart matrices with application to goodness-of-fit testingManuscript (preprint) (Other academic)

The distribution of the vector of the normalized traces of $\small \frac{1}{n}XX'$ and of $\small \Big(\frac{1}{n}XX'\Big)^2$, where the matrix $\small X:p\times n$ follows a matrix normal distribution  and is proved, under the Kolmogorov condition $\small \frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$, to be multivariate normally distributed. Asymptotic moments and cumulants are obtained using a recursive formula derived in  Pielaszkiewicz et al. (2015). We use this result to test for identity of the covariance matrix using a goodness–of–fit approach. The test performs well regarding the power compared to presented alternatives, for both c < 1 or c ≥ 1.

• 45.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE-750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Recursive formula for E(∏i Tr{(WΣ-1)mi}), where W~Wp(∑; n) in finite and asymptotic regime2015Report (Other academic)

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.

• 46.
Department of Energy and Technology, Swedish University of Agriculture Sciences, Uppsala.
On Variance Estimator of Partial Least Squares Regression2010Report (Other academic)

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 and in the

paper 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

rst step the maximum likelihood approach will be applied to a speci c

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 is included in a Krylov structured design matrix describing the

mean.

• 47.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Department of Statistics, Stockholm University.
Explicit estimators in unbalanced mixed linear models 2015In: Festschrift in honor of professor Ghazi Shukur on the occasion of his 60th birthday / [ed] Thomas Holgersson, Växjö: Linnaeus University Press, Växjö, Sweden , 2015, p. 121-125Chapter in book (Other academic)
• 48.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. 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 models2017In: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 26, no 4, p. 299-310Article in journal (Refereed)

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.

• 49.
Department of Statistics, Stockholm University.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences.
Bilinear regression with random effects and reduced rank restrictions2018Report (Other academic)

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

• 50.
Department of Statistics, Stockholm University.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences.
Bilinear regression with rank restrictions on the mean and the dispersion matrix2018Report (Other academic)

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.

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