substantial bias for estimating the uncertainty of parameters. The case bootstrap and three bootstrap methods where both random **effects** and residuals were resampled remained the best methods and selected as bootstrap candidates for **linear** **mixed**-**effects** **models**. The purpose of this work was not to determine which was the best method overall, but to eliminate boot- strap methods that do not perform well even with **linear** **mixed**-**effects** **models**. We did note that the global residual bootstrap was slightly better than individual residual bootstrap in the sparse and large error designs, especially in estimating σ ; which is consistent with the **non**- correlated structure of residuals. In addition, the distribution of resampled residuals obtained by the global residual bootstrap was slightly closer to the original distribution of residuals. The parametric bootstrap performed best across three evaluated designs, but it requires the strongest assumptions (good prior knowledge about model structure and distributions of pa- rameters). If the model is misspecified and the assumptions of normality of random **effects** and residuals are not met, this method may not be robust. In practice, one of the main reasons for using bootstrap is the uncertainty of distribution assumption, the nonparametric bootstrap may therefore preferable to the parametric bootstrap in most applications [9].

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Place Eug`ene Bataillon, 34095 Montpellier Cedex 5, France,
Christian.Lavergne@math.univ-montp2.fr, trottier@math.univ-montp2.fr
Abstract. We address the estimation of Markov (and semi-Markov) switching **linear** **mixed** **models** i.e. **models** that combine **linear** **mixed** **models** with individual- wise random **effects** in a (semi-)Markovian manner. A MCEM-like algorithm whose iterations decompose into three steps (sampling of state sequences given random **effects**, prediction of random **effects** given the state sequence and maximization) is proposed. This statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks.

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Key Words: Diseases and Production, Structural Equation Model, Simultane- ity
27 Standard errors of solutions in large scale **mixed** **models**, applica- tion to **linear** and curvilinear **effects** of inbreeding on production traits. N. Gengler* 1,2 and C. Croquet 1,2 , 1 National Fund for Scientific Research, Brus-

[Figure 3 about here.] [Figure 4 about here.]
6 Discussion
The main original element of this study is the development of the SAEM algorithm for two- levels **non**-**linear** **mixed** **effects** **models**. We extend the SAEM algorithm developed by Kuhn and Lavielle (16), which was not yet adapted to the case of MNLMEMs with two levels of random **effects**. This algorithm will be implemented in the 3.1 version of the monolix software, freely available on the following website: http://monolix.org. The two levels of random **effects** are the between-subject variance and the within-subject (or between-unit) variance, with N subjects and K units, with no restriction on N or K. We show that the SAEM algorithm is split into two parts: an explicit EM algorithm and a stochastic EM part. The integration of the term p(b|φ; θ) in the likelihood results in the derivation of two additional sufficient statistics compared to the original algorithm. Furthermore it uses two intermediate quantities, the conditional expectations and variance of the between-subject random **effects** parameters b. The addition of higher levels of variability would therefore require other extensions of the algorithm.

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KEYWORDS: Likelihood ratio test; Wald test; longitudinal data; **Non**-**linear** **mixed** **effects** **models**; SAEM algorithm; sample size.
1 Introduction
Most clinical trials aim at comparing the efficacy of two different treatments or studying the effect of co-medication or physiological covariates. To assess whether the effect of such covariates implies a better reduction of the disease than without the covariates, several biological endpoints are repeatedly measured along the trial extent. The statistical approaches commonly used to study the influence of the covariate are classically based on the final measurements of this longitudinal data. Alternative methods to improve information extraction from longitudinal studies are analyses based on **linear** or **non**-**linear** **mixed**-**effects** **models** (NLMEMs). Such **models** have been developed for disease evolution studies, to determine the efficacy of anti-viral treatments in human immunodeficiency virus (HIV) [1, 2, 3, 4] or hepatitis B virus [5] infections evaluated through measures of viral load evolution, or prostate cancer treatment assessed by prostate-specific antigen dosage [6]. NLMEMs are also used to model the evolution of functional markers, for instance, for the decay of functional capacity in patients with rheumatoid arthritis [7], or the evolution of the ventilation function in patients with asthma [8]. NLMEMs are also powerful tools to analyze the pharmacokinetic properties of a drug. They allow for decreasing the number of samples per subject, which is an important advantage for interaction studies of protease inhibitors in HIV infected patients, for example [9].

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

of individual-wise random **effects** or by the **linear** **mixed** model (2) in the case of individual-state-wise random **effects**. Since covariates and random **effects** are incorporated in the output process, the successive observations for an individ- ual are assumed to be conditionally independent given the **non**-observable states and the random **effects**. The proposed MCEM-like algorithm can therefore be directly transposed to semi-Markov switching **linear** **mixed** **models**. Given the random **effects**, the state sequences are sampled using the forward-backward algorithm adapted to hidden semi-Markov chains (see Gu´edon (2007) and ref- erences therein). Given a state sequence, the random **effects** are predicted as previously described. The underlying semi-Markov chain parameters (initial probabilities, transition probabilities and state occupancy distributions) and the **linear** **mixed** model parameters (fixed effect parameters, random variance and residual variance) are obtained by maximizing the Monte Carlo approxi- mation of the conditional expectation of the complete-data log-likelihood. The reestimation of the initial probabilities, the transition probabilities and the state occupancy distributions (M-step of the MCEM algorithm) is similar to the rees- timation in the hidden semi-Markov chain case derived by Gu´edon (2003), the smoothed probabilities being simply replaced by counting.

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The work presented here focusses on crossover PK trials analysed by NLMEM. Before the
modelling step, data needs to be collected and we have consequently to define an appropriate de-
sign , which consists of determining a balance between the number of subjects and the number of
samples per subject as well as the allocation of sampling times according to experimental condi- tions. The choice of design has an important impact on the study results, on the precision of the parameter estimates and on the power of the tests [15, 16, 17]. Indeed, a bad choice of design can lead to results which are difficult to interpret and minimise the interest of the study. The main ap- proach for design evaluation has been for a long time based on simulations but it is a cumbersome method, and thus the number of designs which can be evaluated is limited. An alternative approach has been described in the general theory of optimum experimental design used for classical **non**- **linear** **models** [18, 19], relying on the inequality of Rao-Cramer which states that the inverse of

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Abstract
Bivariate **linear** **mixed** **models** are useful when analyzing longitudinal data of two associated markers. In this paper, we present a bivariate **linear** **mixed** model including random **effects** or first-order auto-regressive process and independent measurement error for both markers. Codes and tricks to fit these **models** using SAS Proc **MIXED** are provided. Limitations of this program are discussed and an example in the field of HIV infection is shown. Despite some limitations, SAS Proc **MIXED** is a useful tool that may be easily extendable to multivariate response in longitudinal studies.

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Besides, regularisation methods have already been developped for GLMM, in which the random **effects** allow to model complex dependence structure. Eliot et al. [ 3 ] proposed to extend the classical ridge regression to **Linear** **Mixed** **Models** (LMM). The Expectation- Maximisation algorithm they suggest includes a new step to find the best shrinkage pa- rameter - in the Generalised Cross-Validation (GCV) sense - at each iteration. More re- cently, Groll and Tutz [ 4 ] proposed an L 1 -penalised algorithm for fitting a high-dimensional

The biomechanical data considered in this paper are obtained from a study carried out to understand the coordination patterns of finger forces produced from different tasks. This data cannot be considered independent because of within-individual repeated measurements, and because of simultaneous finger measurements. To fit these data, we propose a methodology focused on **linear** **mixed** **models**. Different random **effects** structures and complex variance- covariance matrices of the error are considered. We highlight how to use the ❧♠❡ R function to deal with such a modelling. The paper is accessible to an audience experienced with **linear** **models**. Some familiarity with the R software is also helpful.

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(2)MAP5, UMR CNRS 8145, Universit´e Paris Descartes, Sorbonne Paris Cit´e, 45 rue des Saints P`eres, 75006 Paris
Abstract
This paper surveys new estimators of the density of a random effect in **linear** **mixed**-**effects** **models**. Data are contaminated by random noise, and we do not observe directly the random effect of interest. The density of the noise is suposed to be known, without assumption on its regularity. However it can also be estimated. We first propose an adaptive nonparametric de- convolution estimation based on a selection method set up in Goldenshluger and Lepski (2011). Then we propose an estimator based on a simpler model selection deviced by contrast penalization. For both of them, **non**-asymptotic L 2 -risk bounds are established implying estimation rates, much better than the expected deconvolution ones. Finally the two data-driven strategies are evaluated on simulations and compared with previous proposals.

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Commodity spaces that are not lattice ordered arise naturally in many economic **models** and the large literature on price decentralization in vector lattices has little, that is obvious, to say in such a setting. An example of such an economic model is portfolio trading when markets are incomplete. It is known that in such **models** all the decentralization results can fail even if the preferences are uniformly proper and the commodity space is finite dimensional. In these **models** consumers are motivated by the payoff of a portfolio. Therefore, the meaningful natural ordering of the port- folio space is the one that compares portfolio payoffs and which is closely related to the notion of first order stochastic domination. In fact, the notion of arbitrage free prices is an order theoretic notion that induces this natural ordering of the portfolio space. Unfortunately, this ordering is rarely a vector lattice ordering when markets are not complete. The basic intuition for this is the following. Generally, when markets are not complete some call and put options cannot be replicated as the payoff of a portfolio of available securities. However, call and put options are closely related to the order structure of the portfolio space. Indeed, every marketed option is a lattice operation in the portfolio space and every lattice operation in the portfolio space is related to what is termed in the finance literature a minimum-cost super replicating portfolio of a call or put option (which need not exist).

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S. Hoffait, G. Kerschen, O. Brüls
LTAS - Department of Aerospace and Mechanical Engineering, Université de Liège, Belgium, {sebastien.hoffait,g.kerschen,o.bruls}@ulg.ac.be
This work addresses the development of robust parameterized reduced-order model (ROM) for **non**- **linear** structures. The Proper Orthogonal Decomposition (POD) approach as well as two methods at- tempting to make it more robust are studied. Their advantages and drawbacks are highlighted on a simple test-case and their applicability for more complex systems is assessed.

In all, this modelling approach is useful to detect, quantify, and hierarchize **effects** of site index, stand density and mixture **effects**, provided that height and diameter are measured (very few required datasets….)
BUT If it gives insight on overall stand production and between tree

Comparison of the estimated Gaussian hidden semi-Markov chain (GHSMC) parameters (i.e. where the influence of covariates and the inter-individual heterogeneity are not taken into account) with the estimated semi-Markov switching **linear** **mixed** model (SMS-LMM) parameters (state occupancy distributions and marginal observation distributions). The regression parameters, the cumulative rainfall effect and the variability decomposition are given

t |˜ x(t + 1) − f (˜ x(t), ˜ u(t))| 2
, or similar, over the unknown parameters of f(·). A similar optimization can be set up for g(·). This is typically very cheap computationally, often reducing to basic least squares. However, if there is no incremental stability requirement then small equation errors do not imply small simulation errors over extended time intervals. For large scale and nonlinear problems it is not unusual to find unstable **models** by this

Robust adaptive beamforming
a b s t r a c t
In parameter estimation, it is common place to design a linearly constrained minimum variance estima- tor (LCMVE) to tackle the problem of estimating an unknown parameter vector in a **linear** regression model. So far, the LCMVE has been mainly studied in the context of stationary constraints in stationary or **non**-stationary environments, giving rise to well-established recursive adaptive implementations when multiple observations are available. In this communication, provided that the additive noise sequence is temporally uncorrelated, we determine the family of **non**-stationary constraints leading to LCMVEs which can be computed according to a predictor/corrector recursion similar to the Kalman Filter. A particularly noteworthy feature of the recursive formulation introduced is to be fully adaptive in the context of se- quential estimation as it allows at each new observation to incorporate or not new constraints.

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24. M. Jiang, and P. Tayebati, “Stable 10 ns, kilowatt peak-power pulse generation from a gain-switched Tm-doped fiber laser,” Opt. Lett. 32(13), 1797–1799 (2007).
1. Introduction
Chalcogenide glasses are known for their large transparency window and their large **non** **linear** optical properties. Indeed, they can be transparent from the visible region up to the infrared, up to 12 to 15 µm, depending on their composition. Another remarkable property of chalcogenide glasses is their strong optical **non** linearity. The **non** **linear** refractive index of sulfur based glasses is over 100 times larger than silica one. The **non** **linear** index of selenium and tellurium based glasses can be more than 1000 times larger than silica one [1,2]. Silica microstructured optical fibers (MOFs) were fabricated as soon as 1973 [3] while chalcogenide ones were drawn only in the last decade. The manufacturing of small core fibers (diameter smaller than 5 µm) can be of great interest to enhance the **non** **linear** optical properties for telecom applications such as signal regeneration [4], for supercontinuum generation [5–7] and conversion to the mid infrared using Raman shifting [8–10]. Conversely power transportation and optical countermeasures in the 3-5 and the 8-12 µm windows require large effective mode area and single mode fibers can be designed to permit the propagation of high power Gaussian laser beams. Single mode fiber can be also used for spatial interferometery in the 4- 12 infrared windows [11]. The first chalcogenide MOF, was made in 2000, but no light propagation was demonstrated [12]. Since then, chalcogenide MOF, with light guidance [7, 13] and small mode area [14] were obtained. The usual method to prepare MOFs is the stack and draw technique. This method comes from the silica technology [15] but optical losses of the chalcogenide MOFs produced using it were still larger than the material losses whatever the wavelength [13, 14]. In 2008, we have demonstrated that most of the optical losses were

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