a geometric characterization of c optimal designs for heteroscedastic regression we consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables the c optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest a geometric characterization of c optimal designs in this context is presented which generalizes the classical result of elfving 1952 for c optimal designs as in elfving s famous characterization c optimal designs can be described as representations of boundary points of a convex set however in the case where there appear parameters of interest in the variance the structure of the elfving set is dierent roughly speaking the elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space the c optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new elfving set the theory is illustrated in several examples including pharmacokinetic models with random eects

on the long time integration of stochastic gradient systems this article addresses the weak convergence of numerical methods for brownian dynamics typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic stepsize h 0 convergence behaviour of the error of finite time averages recently it has been demonstrated by study of fokker planck operators that a non markovian numerical method generates approximations in the long time limit with higher accuracy order second order than would be expected from its weak convergence analysis finite time averages are first order accurate in this article we describe the transition from the transient to the steady state regime of this numerical method by estimating the time dependency of the coefficients in an asymptotic expansion for the weak error demonstrating that the convergence to second order is exponentially rapid in time moreover we provide numerical tests of the theory including comparisons of the efficiencies of the euler maruyama method the popular second order heun method and the non markovian method

curvature adjusted optimal design of sampling times for the inference of pharmacokinetic compartment models in pharmacokinetics compartment models are often used to describe the time course of blood concentration after the administration of a drug in this article we propose an optimal design criterion for precise estimation of parameters included in the compartment model and illustrate the non sequential design of sampling times of blood drug concentration data in individual pharmacokinetics the proposed optimal design criterion minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated by the curvature array therefore the proposed criterion considers the intrinsic and parameter effects nonlinearity underlying the compartment model and so is applicable in a pharmacokinetic experiment where the sample size of the blood drug concentration data is quite small

weak backward error analysis for sdes we consider long time numerical approximations of stochastic differential equations sdes by the euler method in the case where the sde is elliptic or hypoelliptic we show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified kolmogorov equation up to high order terms with respect to the stepsize this implies that every invariant measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion moreover we prove that up to negligible terms the dynamic associated with the euler scheme is exponentially mixing

maximin designs for exponential growth models and heteroscedastic polynomial models this paper is concerned with nonsequential optimal designs for a class of nonlinear growth models which includes the asymptotic regression model this design problem is intimately related to the problem of finding optimal designs for polynomial regression models with only partially known heteroscedastic structure in each case a straightforward application of the usual d optimality criterion would lead to designs which depend on the unknown underlying parameters to overcome this undesirable dependence a maximin approach is adopted the theorem of perron and frobenius on primitive matrices plays a crucial role in the analysis

on the existence and the applications of modified equations for stochastic differential equations in this paper we describe a general framework for deriving modified equations for stochastic differential equations sdes with respect to weak convergence modified equations are derived for a variety of numerical methods such as the euler or the milstein method existence of higher order modified equations is also discussed in the case of linear sdes using the gaussianity of the underlying solutions we derive an sde which the numerical method solves exactly in the weak sense applications of modified equations in the numerical study of langevin equations is also discussed

a note on the de la garza phenomenon for locally optimal designs the celebrated de la garza phenomenon states that for a polynomial regression model of degree p 1 any optimal design can be based on at most p design points in a remarkable paper yang ann statist 38 2010 2499 2524 showed that this phenomenon exists in many locally optimal design problems for nonlinear models in the present note we present a different view point on these findings using results about moment theory and chebyshev systems in particular we show that this phenomenon occurs in an even larger class of models than considered so far

combining multiple comparisons and modeling techniques in dose response studies the analysis of data from dose response studies has long been divided according to two major strategies multiple comparison procedures and model based approaches model based approaches assume a functional relationship between the response and the dose taken as a quantitative factor according to a prespecified parametric model the fitted model is then used to estimate an adequate dose to achieve a desired response but the validity of its conclusions will highly depend on the correct choice of the a priori unknown dose response model multiple comparison procedures regard the dose as a qualitative factor and make very few if any assumptions about the underlying dose response model the primary goal is often to identify the minimum effective dose that is statistically significant and produces a relevant biological effect one approach is to evaluate the significance of contrasts between different dose levels while preserving the family wise error rate such procedures are relatively robust but inference is confined to the selection of the target dose among the dose levels under investigation we describe a unified strategy to the analysis of data from dose response studies which combines multiple comparison and modeling techniques we assume the existence of several candidate parametric models and use multiple comparison techniques to choose the one most likely to represent the true underlying dose response curve while preserving the family wise error rate the selected model is then used to provide inference on adequate doses

a perturbation theory for ergodic markov chains and application to numerical approximations perturbations to markov chains and markov processes are considered the unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of foster lyapunov drift conditions the perturbations are assumed to be uniform in a weak sense on bounded time intervals the long time behavior of the perturbed chain is studied applications are given to numerical approximations of a randomly impulsed ode an ito stochastic differential equation sde and a parabolic stochastic partial differential equation spde subject to space time brownian noise existing perturbation theories for geometrically ergodic markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

optimal designs for michaelis menten kinetic studies many reactions in enzymology are governed by the michaelis menten equation characterising these reactions requires the estimation of the parameters k m and v m a x which determine the michaelis menten equation and this is done by observing rates of reactions at a set of substrate concentrations the choice of substrate concentrations is investigated by determining bayesian d optimal designs for a model in which residuals have a normal distribution with constant variance designs which focus on alternative quantities such as k m or the ratio v m a x k m are also considered the effect on the optimal designs of alternative error distributions is also considered

a method for designing nonlinear univariate calibration a method is proposed for designing nonlinear univariate calibration of measuring instruments the problem addressed is how to select a set of design points standards or calibrators to minimize the errors in the inverse predictions the curve parameters are assumed to vary randomly between calibrations with known expected value and known covariance matrix a design criterion is suggested for analytical procedures according to which the coefficient of variation and the area under the precision profile are minimized

on the de la garza phenomenon deriving optimal designs for nonlinear models is in general challenging one crucial step is to determine the number of support points needed current tools handle this on a case by case basis each combination of model optimality criterion and objective requires its own proof the celebrated de la garza phenomenon states that under a p 1 th degree polynomial regression model any optimal design can be based on at most p design points the minimum number of support points such that all parameters are estimable does this conclusion also hold for nonlinear models if the answer is yes it would be relatively easy to derive any optimal design analytically or numerically in this paper a novel approach is developed to address this question using this new approach it can be easily shown that the de la garza phenomenon exists for many commonly studied nonlinear models such as the emax model exponential model three and four parameter log linear models emax pk1 model as well as many classical polynomial regression models the proposed approach unifies and extends many well known results in the optimal design literature it has four advantages over current tools i it can be applied to many forms of nonlinear models to continuous or discrete data to data with homogeneous or nonhomogeneous errors ii it can be applied to any design region iii it can be applied to multiple stage optimal design and iv it can be easily implemented

optimal designs for estimating individual coefficients in fourier regression models in the common trigonometric regression model we investigate the optimal design problem for the estimation of the individual coefficients where the explanatory variable varies in the interval a a 0 a le pi it is demonstrated that the structure of the optimal design depends sensitively on the size of the design space for many important cases optimal designs can be found explicitly where the complexity of the solution depends on the value of the parameter a and the order of the term for which the corresponding coefficient has to be estimated the main tool of our approach is the reduction of the problem for the trigonometric regression model to a design problem for a polynomial regression in particular we determine the optimal designs for estimating the parameters corresponding to the cosine terms explicitly if the design space is sufficiently small and prove that under this condition all optimal designs for estimating the parameters corresponding to the sine terms are supported at the same points

a geometric characterization of optimal designs for regression models with correlated observations we consider the problem of optimal design of experiments for random effects models especially population models where a small number of correlated observations can be taken on each individual whereas the observations corresponding to different individuals are assumed to be uncorrelated we focus on c optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of elfving from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients the theory is demonstrated by finding optimal designs for a linear model with correlated observations and a non linear random effects population model which is commonly used in pharmaco kinetics

optimal design for goodness of fit of the michaelis menten enzyme kinetic function we construct efficient designs for the michaelis menten enzyme kinetic model capable of checking model assumptions an extended model called emax is also considered for this purpose this model is widely used in pharmacokinetics and reduces to the michaelis menten model for a specific choice of parameter settings our strategy is to find efficient designs for estimating the parameters in the emax model and at the same time test the validity of the michaelis menten model against the emax model by maximizing a minimum of the d or d1 efficiencies taken over a range of values for the nonlinear parameters in particular we show that such designs are a efficient for estimating parameters in the emax model b about 70 efficient for estimating parameters in the michaelis menten model c efficient for testing the michaelis menten model against the emax model and d robust with respect to misspecification of the unknown parameters in the nonlinear model

bayesian t optimal discriminating designs the problem of constructing bayesian optimal discriminating designs for a class of regression models with respect to the t optimality criterion introduced by atkinson and fedorov 1975a is considered it is demonstrated that the discretization of the integral with respect to the prior distribution leads to locally t optimal discriminating design problems with a large number of model comparisons current methodology for the numerical construction of discrimination designs can only deal with a few comparisons but the discretization of the bayesian prior easily yields to discrimination design problems for more than 100 competing models a new efficient method is developed to deal with problems of this type it combines some features of the classical exchange type algorithm with the gradient methods convergence is proved and it is demonstrated that the new method can find bayesian optimal discriminating designs in situations where all currently available procedures fail

postprocessed integrators for the high order integration of ergodic sdes the concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times the idea is to enhance the accuracy of a numerical method by using an appropriate change of variables called the processor we show that this technique can be extended to the stochastic context for the construction of new high order integrators for the sampling of the invariant measure of ergodic systems the approach is illustrated with modifications of the stochastic theta method applied to brownian dynamics where postprocessors achieving order two are introduced numerical experiments including stiff ergodic systems illustrate the efficiency and versatility of the approach

characterization of c l and ϕk optimal designs for a class of non linear multiple regression models optimal designs for multiple regression models are determined we consider a general class of non linear models including proportional hazards models with different censoring schemes the poisson and the negative binomial model for these models we provide a complete characterization of c optimal designs for all vectors c in the case of a single covariate for multiple regression with an arbitrary number of covariates c optimal designs for certain vectors c are derived analytically using some general results on the structure of optimal designs for multiple regression we determine l and k optimal designs for models with an arbitrary number of covariates

long time accuracy of lie trotter splitting methods for langevin dynamics a new characterization of sufficient conditions for the lie trotter splitting to capture the numerical invariant measure of nonlinear ergodic langevin dynamics up to an arbitrary order is discussed our characterization relies on backward error analysis and needs weaker assumptions than assumed so far in the literature in particular neither high weak order of the splitting scheme nor symplecticity are necessary to achieve high order approximation of the invariant measure of the langevin dynamics numerical experiments confirm our theoretical findings

optimal designs for frequentist model averaging we consider the problem of designing experiments for estimating a target parameter in regression analysis when there is uncertainty about the parametric form of the regression function a new optimality criterion is proposed that chooses the experimental design to minimize the asymptotic mean squared error of the frequentist model averaging estimate necessary conditions for the optimal solution of a locally and bayesian optimal design problem are established the results are illustrated in several examples and it is demonstrated that bayesian optimal designs can yield a reduction of the mean squared error of the model averaging estimator by up to 45

further characterizations of design optimality and admissibility for partial parameter estimation in linear regression the paper gives a contribution to the problem of finding optimal linear regression designs when only s out of k regression parameters are to be estimated also a treatment of design admissibility for the parameters of interest is included previous results of kiefer and wolfowitz 1959 karlin and studden 1966 and atwood 1969 are generalized in particular a connection to tchebycheff type approximation of ri valued functions is found which has been known in case s 1 strengthened versions of the results are obtained for invariant designs in situations when invariance properties of the regression setup are available applications are given to multiple quadratic regression and to one dimensional polynomial regression 1 introduction consider a linear regression y x a f x x e with an unknown coefficient vector a e r k and a known r valued function f on the experimental region y the range f y is assumed to be compact as usual observations y1 yn on the dependent variable y taken at points xl xn e e x respectively are assumed to have expectations y x 1 0 and to be pairwise uncorrelated suppose that the experiment aims at estimating ka where k is a given s x k matrix of rank s for example when the first s components of a are to be estimated one has k is 0 where it denotes the unit matrix of order s we will study design optimality and admissibility for estimating ka within the approximate design theory an approximate design is a probability measure on x with finite support the infornation matrix per observation and unit of variance of f is m t ff x f x dd x e f x f x ox f x supp

projective modules for the symmetric group and young s seminormal form abstract we study the representation theory of the symmetric group s n in positive characteristic p using features of the llt algorithm we give a conjectural description of the projective cover p of the simple module d where is a p restricted partition such that all ladders of the corresponding ladder partition are of order less than p inspired by the recent theory of klr algebras we explain an algorithm that allows us to verify this conjectural description for n 15 at least

optimal designs for dose finding studies with an active control summary dose finding studies often compare several doses of a new compound with a marketed standard treatment as an active control in the past however research has focused mostly on experimental designs for placebo controlled dose finding studies to the best of our knowledge optimal designs for dose finding studies with an active control have not been considered so far as the statistical analysis for an active controlled dose finding study can be formulated in terms of a mixture of two regression models the related design problem is different from what has been investigated before in the literature we present a rigorous approach to the problem of determining optimal designs for estimating the smallest dose achieving the same treatment effect as the active control we determine explicitly the locally optimal designs for a broad class of models employed in such studies we also discuss robust design strategies and determine related bayesian and standardized minimax optimal designs we illustrate the results by investigating alternative designs for a clinical trial which has recently appeared in a consulting project of one of the authors

numerical integration of stochastic differential equations with nonglobally lipschitz coefficients we propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations sdes with nonglobally lipschitz coefficients following this concept we discard the approximate trajectories which leave a sufficiently large sphere we prove that accuracy of any method of weak order p is estimated by varepsilon o h p where varepsilon can be made arbitrarily small with increasing radius of the sphere the results obtained are supported by numerical experiments

long run accuracy of variational integrators in the stochastic context this paper presents a lie trotter splitting for inertial langevin equations geometric langevin algorithm and analyzes its long time statistical properties the splitting is defined as a composition of a variational integrator with an ornstein uhlenbeck flow assuming that the exact solution and the splitting are geometrically ergodic the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial langevin equations to within the accuracy of the variational integrator in representing the hamiltonian in particular if the variational integrator admits no energy error then the method samples the invariant measure of inertial langevin equations without error numerical validation is provided using explicit variational integrators with first second and fourth order accuracy

on the efficiency of two stage response adaptive designs in this paper we investigate the efficiency of response adaptive locally optimum designs we focus on two stage adaptive designs where after the first stage the accrued data are used to determine a locally optimum design for the second stage on the basis of an explicit expansion of the information matrix we compare the variance of the maximum likelihood estimates obtained from a two stage adaptive design and a fixed design without adaptation for several one parameter models we provide explicit expressions for the relative efficiency of these two designs which is seen to depend sensitively on the statistical problem under investigation in particular we show that in non linear regression models with moderate or large variances the first stage sample size of an adaptive design should be chosen sufficiently large in order to address variability in the interim parameter estimates these findings support the results of recent simulation studies conducted to compare adaptive designs in more complex situations we finally present an application to a real clinical dose finding trial aiming at the estimation of the smallest dose achieving a certain percentage of the maximum treatment effect by using a three parameteremax model copyright 2012 john wiley sons ltd

optimal designs for quantile regression models despite their importance optimal designs for quantile regression models have not been developed so far in this article we investigate the d optimal design problem for nonlinear quantile regression analysis we provide a necessary condition to check the optimality of a given design and use it to determine bounds for the number of support points of locally d optimal designs the results are illustrated determining locally bayesian and standardized maximin d optimal designs for quantile regression analysis in the michaelis menten and emax model which are widely used in such important fields as toxicology pharmacokinetics and dose response modeling

grading the translation functors in type a we deal with the representation theory of quantum groups and hecke algebras at roots of unity we relate the philosophy of andersen jantzen and soergel on graded translated functors to the lascoux leclerc and thibon algorithm this goes via the murphy standard basis theory and the idempotents coming from the murphy jucys operators our results lead to a guess on a tilting algorithm outside the lowest p2 alcove which at least in the sl2 case coincides with erdmann s results

high order integrator for sampling the invariant distribution of a class of parabolic stochastic pdes with additive space time noise we introduce a time integrator to sample with high order of accuracy the invariant distribution for a class of semilinear spdes driven by an additive space time noise combined with a postprocessor the new method is a modification with negligible overhead of the standard linearized implicit euler maruyama method we first provide an analysis of the integrator when applied for sdes finite dimension where we prove that the method has order 2 for the approximation of the invariant distribution instead of 1 we then perform a stability analysis of the integrator in the semilinear spde context and we prove in a linear case that a higher order of convergence is achieved numerical experiments including the semilinear heat equation driven by space time white noise confirm the theoretical findings and illustrate the efficiency of the approach

optimal designs for the emax log linear and exponential models we derive locally d and edp optimal designs for the exponential log linear and three parameter emax models for each model the locally d and ed p optimal designs are supported at the same set of points while the corresponding weights are different this indicates that for a given model d optimal designs are efficient for estimating the smallest dose that achieves 100p of the maximum effect in the observed dose range conversely ed p optimal designs also yield good d efficiencies we illustrate the results using several examples and demonstrate that locally d and ed p optimal designs for the emax log linear and exponential models are relatively robust with respect to misspecification of the model parameters

optimal designs for identifying the degree of a polynomial regression if an experimenter wants to determine the degree of a polynomial regression on the basis of a sample of observations anderson showed that the following method is optimal starting with the highest specified degree the procedure is to test in sequence whether the coefficients are 0 in this paper optimal designs for anderson s procedure are determined explicitly the optimal design maximizes the minimum power of a given set of alternatives

some robust type d optimal designs in polynomial regression abstract consider a polynomial regression situation on an interval and also a robustness type formulation of stigler s a technique involving canonical moments is discussed and some explicit solutions are given

t optimum designs for discrimination between two multiresponse dynamic models summary the paper is concerned with a problem of finding an optimum experimental design for discriminating between two rival multiresponse models the criterion of optimality that we use is based on the sum of squares of deviations between the models and picks up the design points for which the divergence is maximum an important part of our criterion is an additional vector of experimental conditions which may affect the design we give the necessary conditions for the design and the additional parameters of the experiment to be optimum we present the algorithm for the numerical optimization procedure and we show the relevance of these methods to dynamic systems especially to chemical kinetic models

high order numerical approximation of the invariant measure of ergodic sdes we introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations independently of the weak order of accuracy of the method we then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic sdes brownian and langevin dynamics with an accuracy independent of the weak order of the underlying method numerical experiments confirm our theoretical findings

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

designing experiments with respect to standardized optimality criteria we introduce a new class of standardized optimality criteria which depend on standardized covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces

invariant measures of the milstein method for stochastic differential equations with commutative noise abstract in this paper the milstein method is used to approximate invariant measures of stochastic differential equations with commutative noise the decay rate of the transition probability kernel generated by the milstein method to the unique invariant measure of the method is observed to be exponential with respect to the time variable the convergence rate of the numerical invariant measure to the underlying counterpart is shown to be one numerical simulations are presented to demonstrate the theoretical results

numerical analysis on ergodic limit of approximations for stochastic nls equation via multi symplectic scheme we consider a finite dimensional approximation of the stochastic nonlinear schrodinger equation driven by multiplicative noise which is derived by applying a symplectic method to the original equation in spatial direction both the unique ergodicity and the charge conservation law for this finite dimensional approximation are obtained on the unit sphere to simulate the ergodic limit over long time for the finite dimensional approximation we discretize it further in the temporal direction to obtain a fully discrete scheme which inherits not only the stochastic multi symplecticity and charge conservation law of the original equation but also the unique ergodicity of the finite dimensional approximation the temporal average of the fully discrete numerical solution is proved to converge to the ergodic limit with order 1 with respect to the time step for a fixed spatial step numerical experiments verify our theoretical results on charge conservation ergodicity and weak convergence

optimal designs for the identification of the order of a fourier regression for the fourier regression model we determine optimal designs for identifying the order of periodicity it is shown that the optimal design problem for trigonometric regression models is equivalent to the problem of optimal design for discriminating between certain homo and heteroscedastic polynomial regression models these optimization problems are then solved using the theory of canonical moments and the optimal discriminating designs for the fourier regression model can be found explicitly in contrast to many other optimality criteria for the trigonometric regression model the optimal discriminating designs are not uniformly distributed on equidistant points

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

compound optimal allocation for individual and collective ethics in binary clinical trials in recent years several authors have investigated response adaptive allocation rules for comparative clinical trials in order to favour at each stage of the trial the treatment that appears to be best in this paper we define admissible allocations namely treatment assignments that cannot be simultaneously improved upon with respect to both a specific design criterion reflecting the inferential properties of the experiment and the proportion of patients assigned to the best treatment or treatments we survey existing designs from this viewpoint we also suggest combining information and ethical considerations by taking a suitable weighted mean of two corresponding standardized criteria with weights that depend on the actual treatment effects this compound criterion leads to a locally optimal allocation that can be targeted by some response adaptive randomization rule the paper mainly deals with the case of two treatments but the suggested methodology is shown to extend to more than two

optimal discriminating designs for several competing regression models the problem of constructing optimal designs for a class of regression models is considered we investigate a version of the tp optimality criterion as introduced by atkinson and fedorov 1975b and demonstrate that optimal designs with respect to this type of criteria can be obtained by solving nonlinear vector valued approximation problems we provide a characterization of the best approximations in this context and use these results to develop an efficient algorithm for the determination of the optimal discriminating designs the results are illustrated by fnumerical examples

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

efficient rounding of approximate designs summary discretization methods to round an approximate design into an exact design for a given sample size n are discussed there is a unique method called efficient rounding which has the smallest loss of efficiency under a wide family of optimality criteria the efficient rounding method is a multiplier method of apportionment which otherwise is known as the method of john quincy adams or the method of smallest divisors

t optimal designs for discrimination between two polynomial models the paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree n 2 and n in a fundamental paper atkinson and fedorov 1975a proposed the t optimality criterion for this purpose recently atkinson 2010 determined t optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide a half of the circle into equal parts if the coefficient of xn 1 in the polynomial of larger degree vanishes in the present paper we give a strong justification of the conjecture and determine all t optimal designs explicitly for any degree n n in particular we show that there exists a one dimensional class of t optimal designs moreover we also present a generalization to the case when the ratio between the coefficients of xn 1 and xn is smaller than a certain critical value because of the complexity of the optimization problem t optimal designs have only been determined numerically so far and this paper provides the first explicit solution of the t optimal design problem since its introduction by atkinson and fedorov 1975a finally for the remaining cases where the ratio of coefficients is larger than the critical value we propose a numerical procedure to calculate the t optimal designs the results are also illustrated in an example ams subject classification 62k05

computing ergodic limits for langevin equations abstract to evaluate averages with respect to the invariant law for langevin type equations one has to integrate a system over long time intervals especially when dissipation is small this is a challenging problem from the computational point of view since coefficients of the langevin equations are typically not globally lipschitz the difficulties are redoubled in the nonglobal lipschitz case we observe an exploding behavior of some approximate trajectories due to many usual numerical methods in g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 we propose and justify a concept which in principle allows us to apply any numerical method to stochastic differential equations with nonglobally lipschitz coefficients here quasi symplectic integrators from g n milstein m v tretyakov quasi symplectic methods for langevin type equations ima j numer anal 23 2003 593 626 they are the most appropriate methods for solving langevin type equations especially when damping is small and the concept of g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 are applied for calculation of ergodic limits special attention is paid to the case when the invariant measure is known e g when this measure is gibbsian some computer experiments with three model problems van der pol s equation with additive noise one dimensional arrays of oscillators in thermal equilibrium and a physical pendulum with linear friction and additive noise are presented

multi objective optimal designs in comparative clinical trials with covariates the reinforced doubly adaptive biased coin design the present paper deals with the problem of allocating patients to two competing treatments in the presence of covariates or prognostic factors in order to achieve a good trade off among ethical concerns inferential precision and randomness in the treatment allocations in particular we suggest a multipurpose design methodology that combines efficiency and ethical gain when the linear homoscedastic model with both treatment covariate interactions and interactions among covariates is adopted the ensuing compound optimal allocations of the treatments depend on the covariates and their distribution on the population of interest as well as on the unknown parameters of the model therefore we introduce the reinforced doubly adaptive biased coin design namely a general class of covariate adjusted response adaptive procedures that includes both continuous and discontinuous randomization functions aimed to target any desired allocation proportion the properties of this proposal are described both theoretically and through simulations

optimal discrimination designs we consider the problem of constructing optimal designs for model discrimination between competing regression models various new properties of optimal designs with respect to the popular t optimality criterion are derived which in many circumstances allow an explicit determination of t optimal designs it is also demonstrated that in nested linear models the number of support points of t optimal designs is usually too small to estimate all parameters in the extended model in many cases t optimal designs are usually not unique and we give a characterization of all t optimal designs finally t optimal designs are compared with optimal discriminating designs with respect to alternative criteria by means of a small simulation study

adaptive thermostats for noisy gradient systems we study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system extended stochastic dynamical methods are discussed which have application to multiscale models nonequilibrium molecular dynamics and bayesian sampling techniques arising in emerging machine learning applications in addition to providing a more comprehensive discussion of the foundations of these methods we propose a new numerical method for the adaptive langevin stochastic gradient nose hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature we also demonstrate that the newly established method inherits a superconvergence property fourth order convergence to the invariant measure for configurational quantities recently demonstrated in the setting of langevin dynamics our findings are verified by numerical experiments

exploration of the non asymptotic bias and variance of stochastic gradient langevin dynamics applying standard markov chain monte carlo mcmc algorithms to large data sets is computationally infeasible the recently proposed stochastic gradient langevin dynamics sgld method circumvents this problem in three ways it generates proposed moves using only a subset of the data it skips the metropolis hastings accept reject step and it uses sequences of decreasing step sizes in teh et al 2014 we provided the mathematical foundations for the decreasing step size sgld including consistency and a central limit theorem however in practice the sgld is run for a relatively small number of iterations and its step size is not decreased to zero the present article investigates the behaviour of the sgld with fixed step size in particular we characterise the asymptotic bias explicitly along with its dependence on the step size and the variance of the stochastic gradient on that basis a modified sgld which removes the asymptotic bias due to the variance of the stochastic gradients up to first order in the step size is derived moreover we are able to obtain bounds on the finite time bias variance and mean squared error mse the theory is illustrated with a gaussian toy model for which the bias and the mse for the estimation of moments can be obtained explicitly for this toy model we study the gain of the sgld over the standard euler method in the limit of large data sets

optimal discrimination designs for semiparametric models author s guchennko roman melas v b wong weng kee abstract much work on optimal discrimination designs assumes that the models of interest are fullyspecified apart from unknown parameters recent work allows errors in the models to be nonnormallydistributed but still requires the specification of the mean structures otsu 2008 proposedoptimal discriminating designs for semiparametric models by generalizing the kullback leibleroptimality criterion proposed by lopez fidalgo et al 2007 this paper develops a relativelysimple strategy for finding an optimal discrimination design we also formulate equivalence theoremsto confirm optimality of a design and derive relations between optimal designs found herefor discriminating semiparametric models and those commonly used in optimal discriminationdesign problems

robust and efficient designs for the michaelis menten model for the michaelis menten model we determine designs that maximize the minimum of the d efficiencies over a certain interval for the nonlinear parameter the best two point designs can be found explicitly and a characterization is given when these designs are optimal within the class of all designs in most cases of practical interest the determined designs are highly efficient and robust with respect to misspecification of the nonlinear parameter the results are illustrated and applied in an example of a hormone receptor assay

optimal designs for the rasch model in this paper optimal designs will be derived for estimating the ability parameters of the rasch model when difficulty parameters are known it is well established that a design is locally d optimal if the ability and difficulty coincide but locally optimal designs require that the ability parameters to be estimated are known to attenuate this very restrictive assumption prior knowledge on the ability parameter may be incorporated within a bayesian approach several symmetric weight distributions e g uniform normal and logistic distributions will be considered furthermore maximin efficient designs are developed where the minimal efficiency is maximized over a specified range of ability parameters

accurate stationary densities with partitioned numerical methods for stochastic differential equations we devise explicit partitioned numerical methods for second order in time scalar stochastic differential equations using one gaussian random variable per timestep the construction proceeds by analysis of the stationary density in the case of constant coefficient linear equations imposing exact stationary statistics in the position variable and absence of correlation between position and velocity the remaining error is in the velocity variable a new two stage reverse leapfrog method has good properties in the position variable and is symplectic in the limit of zero damping explicit new runge kutta leapfrog methods are constructed sharing the property that q n 1 q n frac 1 2 p n p n 1 delta t whose mean square velocity order increases with the number of stages

error estimates on ergodic properties of discretized feynman kac semigroups we consider the numerical analysis of the time discretization of feynman kac semigroups associated with diffusion processes these semigroups naturally appear in several fields such as large deviation theory diffusion monte carlo or non linear filtering we present errors estimates a la talay tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized as well as on the leading eigenvalue of the generator of the dynamics which corresponds to the rate of creation of probability this provides criteria to construct efficient integration schemes of feynman kac dynamics as well as a mathematical justification of numerical results already observed in the diffusion monte carlo community our analysis is illustrated by numerical simulations

robust t optimal discriminating designs this paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the t optimality criterion introduced by atkinson and fedorov 1975a t optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification as a solution of this problem we propose a bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the t optimality criterion it is shown that the corresponding bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria for the problem of discriminating between two polynomial regression models which differ in the degree by two the robust t optimal discriminating designs can be found explicitly the results are illustrated in several examples ams subject classification 62k05