buckling of stochastically heterogeneous beams using a functional perturbation method abstract the buckling load and its probabilistic nature average and variance of bernoulli beams with stochastic material bending stiffness properties is derived analytically by a new functional perturbation method fpm a buckling shape function is assumed based on the homogeneous solution and additional terms to account for the morphology effects the buckling load in the transcendental equation is treated as a functional of the bending modulus stiffness or compliance field applying a functional perturbation to the above equation the buckling load is found analytically to any desired degree of accuracy as a function of material morphology the fpm is executed using both stiffness and compliance statistical data the impact of each of the two data sources on the solution accuracy is examined showing that compliance based solutions are accurate for small correlation lengths statically indeterminate problems can be treated with no additional effort an example of a simply supported beam is solved in detail comparison with previous studies where stochastic finite element and monte carlo simulation were used showed the relative accuracy and insight capabilities of the method the clamped free case is also studied to demonstrate that symmetry conditions used for homogeneous beams to find the buckling load on the basis of a simply supported case are not valid for heterogeneous beams

effect of strength modulus correlation on reliability of randomly heterogeneous beams abstract the strength reliability of linearly elastic brittle stochastically heterogeneous beams is studied on the basis of the weakest link approach the analysis is formulated by a functional perturbation method resulting in an analytical solution of the failure probability of the beam heterogeneity material morphology is random and confined to longitudinal direction only under bernoulli assumptions the problem is statically indeterminate and external loads are not random the stress field is random and functionally dependent on morphology in particular local strength is also a function of modulus therefore the strength reliability of the beam is morphology dependent both through static indeterminacy and local strength modulus correlation the above is also coupled with the probabilistic nature of strength associated with surface defects and irregularities the case of single indeterminacy clamped simply supported beam is investigated for simplicity it is shown that the strength of the beam is significantly affected by material morphology and that the effect can be either positive increased strength or negative depending on the strength moduli correlation for example for an effective grain size of l 10 and a compliance statistical variance of 1 12 the morphology effect on the allowable design load was found to be in the order of 10 calculation of size effect corresponding to strength showed a complex non classical grain size dependency

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

statistical modeling of heterogeneous micro beams statistical characteristics averages and variances of heterogeneous small size beams micro beams composed of micro elements grains of randomly distributed stiffness isotropic is studied only longitudinal variations are considered element to beam size length ratio is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient using standard statistical tools probability density and correlation functions it is shown that for indeterminate cases even average deflections obtained explicitly are load dependent and are not identical with their corresponding homogeneous case bounds for the statistical dispersion of deflections and reaction forces are found analytically for some cases of weak heterogeneity it is shown that dispersion parameters for indeterminate problems can be obtained by superposition using both averages and dispersion data of corresponding determinate cases leading to reciprocal type relations

bayesian estimation and experimental design in linear regression models estimation and design as a bayesian decision problem choice of a prior distribution conjugate prior distributions bayes estimation of the regression parameter optimality and robustness of the bayes estimator bayesian interpretation of estimators using non bayesian prior knowledge bayes estimation in case of prior ignorance further problems the design problem for the linear bayes estimator characterization of optimal designs construction of optimal continuous designs construction of exact optimal designs

on hermite interpolation by cauchy vandermonde systems the lagrange formula the adjoint and the inverse of a cauchy vandermonde matrix for a given cauchy vandermonde system and for given multiple nodes a lagrange type formula for the interpolant is derived interpolating a given function in the sense of hermite we give explicit analytic representations of the basic functions in terms of the nodes and prescribed poles they are used to derive formulas for the entries of the adjoint of the confluent cauchy vandermonde matrix corresponding to the interpolation problem thus providing an explicit representation of its inverse

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

optimized functional perturbation method and morphology based effective properties of randomly heterogeneous beams abstract a functional perturbation method fpm has been recently developed for the analysis of stochastically heterogeneous structures for which the heterogeneity scale is not negligible relative to the macro dimensions the fpm is based on considering the target function here the buckling load p as a functional of the stochastic morphology the target function is written as a functional perturbation series near a convenient homogeneous property usually stiffness k or compliance s thus the accuracy depends on the choice of the property around which the perturbation is carried out an optimized fpm ofpm is presented here which concentrates on finding a property k which is a function of k or s such that the target function converges faster this is accomplished by looking for k which minimizes or nulls if possible the second term in the functional perturbation series besides its improved accuracy has also a dual meaning which is related to the notion of effective property however the effectiveness is weak since the property is not purely material but depends on external loading shapes an example of a buckling problem is examined in detail for which is found analytically as a simple power of k which directly depends on morphology comparing the new ofpm with previous fpm and numerical monte carlo finite element results shows the desired improved accuracy the advantages of the ofpm are then shown and discussed

bayesian design for dose response curves with penalized risk this paper considers some bayesian design problems in quantal response analysis an experimenter must choose a set of dose levels and number of independent observations to take at these levels subject to a total sample size in order to estimate some characteristic phi e g ed50 of a tolerance distribution f theta where theta is the vector of unknown parameters it is shown for the logistic model that the sampling variability of the posterior variance of phi is such that the predicted posterior variance alone is an undesirable criterion for design selection a family of penalty functions is introduced that penalizes any excess in the posterior variance over the expected or predicted variance and protects against unexpected outcomes the goal is to find a design that avoids experimental results with little information at the expense of a small sacrifice in the bayes risk numerical results indicate that the chance of an extreme posterior variance can be reduced by sacrificing a small amount of posterior risk

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

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 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

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

optimal designs for a class of nonlinear regression models for a broad class of nonlinear regression models we investigate the local e and c optimal design problem it is demonstrated that in many cases the optimal designs with respect to these optimality criteria are supported at the chebyshev points which are the local extrema of the equi oscillating best approximation of the function f 0 0 by a normalized linear combination of the regression functions in the corresponding linearized model the class of models includes rational logistic and exponential models and for the rational regression models the e and c optimal design problem is solved explicitly in many cases

uncertain buckling its past present and future an authoritative review on stochastic buckling of structures was written by amazigo some quarter century ago the present review summarizes some of the developments which took place since then a brief overview of the effect of uncertainty in the initial geometric imperfections elastic moduli applied forces and thickness variation is given for the benefit of the thoughtful reader the review is of critical nature

self attracting diffusions case of the constant interaction in this paper we study a self attracting diffusion in the case of a constant self attraction and for dimension larger than two we prove that this process converges almost surely

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

bayesian approach to the analysis of nonlinear models implementation and evaluation a traditional approach to parameter estimation and hypothesis testing in nonlinear models is based on least squares procedures error analysis depends on large sample theory bayesian analysis is an alternative approach that avoids substantial errors which could result frorn this dependence this communication is concerned with the implementation of the bayesian approach as an alternative to least squares nonlinear regression special attention is given to the numerical evaluation of multiple integrals and to the behavior of the parameter estimators and their estimated covariances the bayesian approach is evaluated in the light of practical as well as theoretical considerations 1 inttoduction the traditional approach to the statistical analysis of nonlinear models is first to use some numerical method to minimize the sum of squares objective function in order to obtain least squares estimators of the parameters draper and smith 1966 pp 267 275 nelder and mead 1965 and then to apply linear regression theory to the linear part of the taylor series approximation of the model expanded about these estimators in order to obtain the asymptotic covariance matrix bard 1974 pp 176 179 the distributions of the estimators obtained in this manner are known only in the limit as the sample size approaches infinity jennrich 1969 hence analyses based on these statistics may be inappropriate for small sample problems such as those arising in pharmacokinetics wagner 1975 an alternative approach to the statistical analysis of nonlinear models is to utilize methods based on bayes s theorem box and tiao 1973 pp 1 73 parameters are regarded as random variables rather than as unknown constants if a nonlinear model with known error distribution is assumed a correct probability analysis follows and asymptotic theory is not involved in this communication a bayesian approach to nonlinear regression is implemented and evaluated particular attention is given to numerical integration methods and the calculation of confidence regions using the posterior distributions

e optimal designs for rational models in this paper experimental designs for a rational model y p x q x e are investigated where p x o 1 x p x p and q x 1 p 1 x p q x q are polynomials and e is a random error two approaches bayesian d optimal and bayesian optimal design for extrapolation are examined the first criterion maximizes the expected increase in shannon information provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator mle of the mean response at an extrapolation point x e corresponding locally optimal designs are also discussed conditions are derived under which a p q 1 point design is a locally d optimal design the bayesian d optimal design is shown to be independent of the parameters in p x and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model the existence and uniqueness of the locally optimal design for extrapolation are proven the number of support points for the locally optimal design for extrapolation is exactly p q 1 these p q 1 design points are proved to be independent of the extrapolation point x e and the parameters in p x the corresponding weights are also independent of the parameters in p x but depend on x e and are not equal