on the lie symmetry algebra of a general ordinary differential equation we give a description of the lie symmetry algebra of a general ordinary differential equation which clarifies certain issues in its practical use in particular we present examples of symmetry algebras for well known equations which illustrate features of this algebra

transmutations l bases and complete families of solutions of the stationary schrodinger equation in the plane an l basis associated to a linear second order ordinary differential operator l is an infinite sequence of functions k k 0 such that l k 0 for k 0 1 l k k k 1 k 2 for k 2 3 and all k satisfy certain prescribed initial conditions we study the transmutation operators related to l in terms of the transformation of powers of the independent variable x x0 k k 0 to the elements of the l basis and establish a precise form of the transmutation operator realizing this transformation we use this transmutation operator to establish a completeness of an infinite system of solutions of the stationary schrodinger equation from a certain class the system of solutions is obtained as an application of the theory of bicomplex pseudoanalytic functions and its completeness was a long sought result its use for constructing reproducing kernels and solving boundary and eigenvalue problems has been considered even without the required completeness justification the obtained result on the completeness opens the way for further development and application of the tools of pseudoanalytic function theory

spectral parameter power series for perturbed bessel equations a spectral parameter power series spps representation for regular solutions of singular bessel type sturm liouville equations with complex coefficients is obtained as well as an spps representation for the entire characteristic function of the corresponding spectral problem on a finite interval it is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the sturm liouville problem based on the spps representation a new mapping property of the transmutation operator for the considered perturbed bessel operator is obtained and a new numerical method for solving corresponding spectral problems is developed the range of applicability of the method includes complex coefficients complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator on a set of known test problems we show that the developed numerical method based on the spps representation is highly competitive in comparison to the best available solvers such as sleign2 matslise and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the spps method delivers excellent numerical results

optimal rates of convergence for deconvolving a density abstract suppose that the sum of two independent random variables x and z is observed where z denotes measurement error and has a known distribution and where the unknown density f of x is to be estimated one application is the estimation of a prior density for a sequence of location parameters a second application arises in the errors in variables problem for nonlinear and generalized linear models when one attempts to model the distribution of the true but unobservable covariates this article shows that if z is normally distributed and f has k bounded derivatives then the fastest attainable convergence rate of any nonparametric estimator of f is only log n k 2 therefore deconvolution with normal errors may not be a practical proposition other error distributions are also treated stefanski carroll 1987a estimators achieve the optimal rates the results given have versions for multiplicative errors where they imply that even optimal rates are exceptionally slow

on parametric bootstrap methods for small area prediction summary the particularly wide range of applications of small area prediction e g in policy making decisions has meant that this topic has received substantial attention in recent years the problems of estimating mean squared predictive error of correcting that estimator for bias and of constructing prediction intervals have been addressed by various workers although existing methodology is still restricted to a narrow range of models to overcome this difficulty we develop new bootstrap based methods which are applicable in very general settings for constructing bias corrected estimators of mean squared error and for computing prediction regions unlike existing techniques which are based largely on taylor expansions our bias corrected mean squared error estimators do not require analytical calculation they also have the property that they are non negative our prediction intervals have a high degree of coverage accuracy o n 3 where n is the number of areas if double bootstrap methods are employed the techniques do not depend on the form of the small area estimator and are applicable to general two level small area models where the variables at either level can be discrete or continuous and in particular can be non normal most importantly the new methods are simple and easy to apply

transmutations for darboux transformed operators with applications we solve the following problem let q1 be a continuous complex valued potential of a stationary schr dinger operator defined on a segment a a and q2 be the potential of a darboux transformed schr dinger operator that is where f is a nonvanishing solution of the equation suppose a transmutation operator t1 is known such that for any u c2 a a find an analogous transmutation operator for it is well known that the transmutation operators can be realized in the form of volterra integral operators with continuously differentiable kernels given a kernel k1 of the transmutation operator t1 we find the kernel k2 of t2 in a closed form in terms of k1 as a corollary interesting commutation relations between t1 and t2 are obtained which then are used in order to construct the transmutation operator for the one dimensional dirac system with a scalar potential

an algorithm for computing the nonparametric mle of a mixing distribution abstract a fast algorithm for calculating the nonparametric maximum likelihood estimate mle of a mixing distribution g in a mixture model is discussed the literature contains several methods for computing the nonparametric mle of g but these methods are slow in this article we develop an algorithm for maximizing the log likelihood l g over the family of all distribution functions that yields the nonparametric mle of g in some semiparametric problems a structural or fixed parameter is of interest and we are interested in computing profile likelihood sup gl g for a grid of values of the algorithm that we propose is fast enough for this purpose examples illustrate and compare the algorithms one taken from an article by laird the common mean problem and one taken from the literature on optimal experimental design it is also noted that routines from the literature on semi infinite programming may be used to compute the profile log likelihood

multiple testing for neuroimaging via hidden markov random field traditional voxel level multiple testing procedures in neuroimaging mostly p value based often ignore the spatial correlations among neighboring voxels and thus suffer from substantial loss of power we extend the local significance index based procedure originally developed for the hidden markov chain models which aims to minimize the false nondiscovery rate subject to a constraint on the false discovery rate to three dimensional neuroimaging data using a hidden markov random field model a generalized expectation maximization algorithm for maximizing the penalized likelihood is proposed for estimating the model parameters extensive simulations show that the proposed approach is more powerful than conventional false discovery rate procedures we apply the method to the comparison between mild cognitive impairment a disease status with increased risk of developing alzheimer s or another dementia and normal controls in the fdg pet imaging study of the alzheimer s disease neuroimaging initiative

estimating species richness by a poisson compound gamma model we propose a poisson compound gamma approach for species richness estimation based on the denseness and nesting properties of the gamma mixture we fix the shape parameter of each gamma component at a unified value and estimate the mixture using nonparametric maximum likelihood a least squares crossvalidation procedure is proposed for the choice of the common shape parameter the performance of the resulting estimator of n is assessed using numerical studies and genomic data

simultaneous testing of grouped hypotheses finding needles in multiple haystacks in large scale multiple testing problems data are often collected from heterogeneous sources and hypotheses form into groups that exhibit different characteristics conventional approaches including the pooled and separate analyses fail to efficiently utilize the external grouping information we develop a compound decision theoretic framework for testing grouped hypotheses and introduce an oracle procedure that minimizes the false nondiscovery rate subject to a constraint on the false discovery rate it is shown that both the pooled and separate analyses can be uniformly improved by the oracle procedure we then propose a data driven procedure that is shown to be asymptotically optimal simulation studies show that our procedures enjoy superior performance and yield the most accurate results in comparison with both the pooled and separate procedures a real data example with grouped hypotheses is studied in detail using different methods both theoretical and numerical results demonstrate that exploit

estimated estimating equations semiparametric inference for clustered and longitudinal data summary we introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions this estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi likelihood regression where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link and variance covariance is an unknown smooth function of the marginal means we propose to estimate the nonparametric link and variance covariance functions via smoothing methods whereas the regression parameters are obtained via the estimated estimating equations these are score equations that contain nonparametric function estimates the proposed estimated estimating equations approach is motivated by its flexibility and easy implementation moreover if data follow a generalized linear mixed model with either a specified or an unspecified distribution of random effects and link function the model proposed emerges as the corresponding marginal population average version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model without the need to specify any other components of this generalized linear mixed model among marginal models the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations applications of estimated estimating equations include diagnostics and link selection the asymptotic distribution of the proposed estimators for the model parameters is derived enabling statistical inference practical illustrations include poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses

weighted nonparametric maximum likelihood estimate of a mixing distribution in nonrandomized clinical trials hierarchical models have a variety of applications including multi center clinical trials local estimation of disease rates longitudinal studies risk assessment and meta analysis in a hierarchical model observations are sampled conditional on individual unit specific parameters and these parameters are sampled from a mixing distribution in observational studies or nonrandomized clinical trails observations may be biased samples from a population and heterogeneous with respect to some confounding factors without controlling the heterogeneity in the sample the standard estimation of the mixing distribution may lead to inaccurate statistical inferences in this article we propose a weighted nonparametric maximum likelihood estimate npmle of the mixing distribution and its smoothed version via weighted smoothing by roughening the proposed estimator reduces bias by assigning a weight to each subject in the sample the weighted npmle is shown to be weighted self consistent and therefore can be easily calculated through a recursive approach simulation studies were conducted to evaluate the performance of the proposed estimator we applied this method to clinical trial data evaluating a new treatment for stress urinary incontinence

an estimation method for the semiparametric mixed effects model summary a semiparametric mixed effects regression model is proposed for the analysis of clustered or longitudinal data with continuous ordinal or binary outcome the common assumption of gaussian random effects is relaxed by using a predictive recursion method newton and zhang 1999 to provide a nonparametric smooth density estimate a new strategy is introduced to accelerate the algorithm parameter estimates are obtained by maximizing the marginal profile likelihood by powell s conjugate direction search method monte carlo results are presented to show that the method can improve the mean squared error of the fixed effects estimators when the random effects distribution is not gaussian the usefulness of visualizing the random effects density itself is illustrated in the analysis of data from the wisconsin sleep survey the proposed estimation procedure is computationally feasible for quite large data sets

a bayesian population model with hierarchical mixture priors applied to blood count data abstract population pharmacokinetic and pharmacodynamic studies require analyzing nonlinear growth curves fit to multiple measurements from study subjects we propose a class of nonlinear population models with nonparametric second stage priors for analyzing such data the proposed models apply a flexible class of mixtures to implement the nonparametric second stage the discussion is based on a pharmacodynamic study involving longitudinal data consisting of hematologic profiles i e blood counts measured over time of cancer patients undergoing chemotherapy we describe a full posterior analysis in a bayesian framework this includes prediction of future observations profiles and end points for new patients estimation of the mean response function for observed individuals and inference on population characteristics the mixture model is specified and given a hyperprior distribution by means of a dirichlet processes prior on the mixing measure estimation is implemented by a combination of various m

random effects models with non parametric priors we discuss the performance of non parametric maximum likelihood npml estimators for the distribution of a univariate random effect in the analysis of longitudinal data for continuous data we analyse generated and real data sets and compare the npml method to those that assume a gaussian random effects distribution and to ordinary least squares for binary outcomes we use generated data to study the moderate and large sample performance of the npml compared with a method based on a gaussian random effect distribution in logistic regression we find that estimated fixed effects are compatible for all approaches but that appropriate standard errors for the npml require adjusting the likelihood based standard errors we conclude that the non parametric approach provides an attractive alternative to gaussian based methods though additional evaluations are necessary before it can be recommended for general use

a fast algorithm for linear complex chebyshev approximations we propose a new algorithm for finding best minimax polynomial approximations in the complex plane the algorithm is the first satisfactory generalization of the well known remez algorithm for real approximations among all available algorithms ours is the only quadratically convergent one numerical examples are presented to illustrate rapid convergence

liouville transformation analytic approximation of transmutation operators and solution of spectral problems a method for solving spectral problems for the sturm liouville equation p v q v r v 0 based on the approximation of the delsarte transmutation operators combined with the liouville transformation is presented the problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients p q and r by a finite linear combination of certain specially constructed functions related to generalized wave polynomials introduced by khmelnytskaya et al 2013 and kravchenko and torba 2015 the method allows one to compute both lower and higher eigendata with an extreme accuracy several necessary results concerning the action of the liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the sturm liouville operator 1 r d d x p d d x q

computer assisted analysis of mixtures c a man statistical algorithms summary this paper presents various algorithmic approaches for computing the maximum likelihood estimator of the mixing distribution of a one parameter family of densities and provides a unifying computeroriented concept for the statistical analysis of unobserved heterogeneity i e observations stemming from different subpopulations in a univariate sample the case with unknown number of population subgroups as well as the case with known number of population subgroups with emphasis on the first is considered in the computer package c a man computer assisted mixture analysis it includes an algorithmic menu with choices of the em algorithm the vertex exchange algorithm a combination of both as well as the vertex direction method to ensure reliable convergence a steplength menu is provided for the three latter methods each achieving monotonicity for the direction of choice c a man has the option to work with restricted support size that is the case when the number of components is known a priori in the latter case the em algorithm is used applications of mixture modelling in medical problems are discussed

ermakov systems of arbitrary order and dimension structure and linearization ermakov systems of arbitrary order and dimension are constructed these inherit an underlying linear structure based on that recently established for the classical ermakov system as an application alignment of a 2 1 dimensional ermakov and integrable ernst system is shown to produce a novel integrable hybrid of a 2 1 dimensional sinh gordon system and of a conventional ermakov system

triple goal estimates in two stage hierarchical models the beauty of the bayesian approach is its ability to structure complicated models inferential goals and analyses to take full advantage of it methods should be linked to an inferential goal via a loss function for example in the two stage compound sampling model the posterior means are optimal under squared error loss however they can perform poorly in estimating the histogram of the parameters or in ranking them triple goal estimates are motivated by the desire to have a set of estimates that produce good ranks a good parameter histogram and good co ordinate specific estimates no set of estimates can simultaneously optimize these three goals and we seek a set that strikes an effective trade off we evaluate and compare three candidate approaches the posterior means the constrained bayes estimates of louis and ghosh and a new approach that optimizes estimation of the histogram and the ranks mathematical and simulation based analyses support the superiority of the new approach and document its excellent performance for the three inferential goals

non parametric inference for clustered binary and count data when only summary information is available summary data in the form of pairs x y where the response y is a count arise in many applications including problems involving stratified or two stage sampling such data are often analysed by using random effects models where the distribution of y conditional on x and on an unobserved random parameter is taken to be either binomial or poisson and the distribution of is connected through a link function to a random effect the latter is sometimes supposed to be normally distributed but that assumption can lead to serious biases if it is not satisfied this difficulty has motivated an extensive literature on non parametric techniques for cases where the random effect distribution is unknown but that methodology requires detailed cluster level data no non parametric techniques are available for instances where only summary data are accessible we show that the random effect distribution is actually non parametrically identifiable from crude summary data and we suggest a sieve method for inference in this case non parametric approaches to both parameter estimation and prediction are introduced and empirical techniques are developed for choosing tuning parameters these methods are shown to outperform their parametric counterparts when the model is misspecified and to perform almost as well as parametric methods when the model is correct

on solving a class of unbalanced ermakov pinney systems we use an intertwining property of linear differential operators to construct the general solution of a type of coupled ermakov pinney system with two distinct frequency terms

bayesian meta analysis for longitudinal data models using multivariate mixture priors summary we propose a class of longitudinal data models with random effects that generalizes currently used models in two important ways first the random effects model is a flexible mixture of multivariate normals accommodating population heterogeneity outliers and nonlinearity in the regression on subject specific covariates second the model includes a hierarchical extension to allow for meta analysis over related studies the random effects distributions are decomposed into one part that is common across all related studies common measure and one part that is specific to each study and that captures the variability intrinsic between patients within the same study both the common measure and the study specific measures are parameterized as mixture of normals models we carry out inference using reversible jump posterior simulation to allow a random number of terms in the mixtures the sampler takes advantage of the small number of entertained models the motivating application is the analysis of two studies carried out by the cancer and leukemia group b calgb in both studies we record for each patient white blood cell counts wbc over time to characterize the toxic effects of treatment the wbcs are modeled through a nonlinear hierarchical model that gathers the information from both studies

conditional estimation for generalized linear models when covariates are subject specific parameters in a mixed model for longitudinal measurements the relationship between a primary endpoint and features of longitudinal profiles of a continuous response is often of interest and a relevant framework is that of a generalized linear model with covariates that are subject specific random effects in a linear mixed model for the longitudinal measurements naive implementation by imputing subject specific effects from individual regression fits yields biased inference and several methods for reducing this bias have been proposed these require a parametric normality assumption on the random effects which may be unrealistic adapting a strategy of stefanski and carroll 1987 biometrika74 703 716 we propose estimators for the generalized linear model parameters that require no assumptions on the random effects and yield consistent inference regardless of the true distribution the methods are illustrated via simulation and by application to a study of bone mineral density in women transitioning to menopause

semiparametric models for longitudinal data with application to cd4 cell numbers in hiv seroconverters summary the paper describes a semiparametric model for longitudinal data which is illustrated by its application to data on the time evolution of cd4 cell numbers in hiv seroconverters the essential ingredients of the model are a parametric linear model for covariate adjustment a nonparametric estimation of a smooth time trend serial correlation between measurements on an individual subject and random measurement error a back fitting algorithm is used in conjunction with a cross validation prescription to fit the model a notable feature in the application is that the onset of hiv infection is associated with a sudden drop in cd4 cells followed by a longer term slower decay the model is also used to estimate an individual s curve by combining his data with the population curve shrinkage toward the population mean trajectory is controlled in a natural way by the estimated covariance structure of the data

the geometry of mixture likelihoods a general theory in this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set the existence support size likelihood equations and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull it is shown using geometric techniques that the estimator exists under quite general conditions with a support size no larger than the number of distinct observations analysis of the convex dual of the likelihood set leads to a dual maximization problem a convergent algorithm is described the defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures sufficient conditions for uniqueness are given part ii will deal with a special theory for exponential family mixtures

linear mixed models with flexible distributions of random effects for longitudinal data normality of random effects is a routine assumption for the linear mixed model but it may be unrealistic obscuring important features of among individual variation we relax this assumption by approximating the random effects density by the seminonparameteric snp representation of gallant and nychka 1987 econometrics 55 363 390 which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior controlled by a user chosen tuning parameter an advantage is that the marginal likelihood may be expressed in closed form so inference may be carried out using standard optimization techniques we demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality and we illustrate the approach via simulation and using longitudinal data from the framingham study

the effects of mixture distribution misspecification when fitting mixed effects logistic models summary mixed effects logistic models are often used to analyze binary response data which have been gathered in clusters or groups responses are assumed to follow a logistic model within clusters with an intercept which varies across clusters according to a specified probability distribution g in this paper we examine the performance of mixed effects logistic regression analysis when a main component of the model the mixture distribution is misspecified we show that when the mixture distribution is misspecified estimates of model parameters including the effects of covariates typically are asymptotically biased i e inconsistent however we present some approximations which suggest that the magnitude of the bias in the estimated covariate effects is typically small these findings are corroborated by a set of simulations which also suggest that valid variance estimates of estimated covariate effects can be obtained when the mixture distribution is misspecified

a general maximum likelihood analysis of variance components in generalized linear models this paper describes an em algorithm for nonparametric maximum likelihood ml estimation in generalized linear models with variance component structure the algorithm provides an alternative analysis to approximate mql and pql analyses mcgilchrist and aisbett 1991 biometrical journal 33 131 141 breslow and clayton 1993 journal of the american statistical association 88 9 25 mcgilchrist 1994 journal of the royal statistical society series b 56 61 69 goldstein 1995 multilevel statistical models and to gee analyses liang and zeger 1986 biometrika 73 13 22 the algorithm first given by hinde and wood 1987 in longitudinal data analysis 110 126 is a generalization of that for random effect models for overdispersion in generalized linear models described in aitkin 1996 statistics and computing 6 251 262 the algorithm is initially derived as a form of gaussian quadrature assuming a normal mixing distribution but with only slight variation it can be used for a completely unknown mixing distribution giving a straightforward method for the fully nonparametric ml estimation of this distribution this is of value because the ml estimates of the glm parameters can be sensitive to the specification of a parametric form for the mixing distribution the nonparametric analysis can be extended straightforwardly to general random parameter models with full npml estimation of the joint distribution of the random parameters this can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters a simple method is described for obtaining correct standard errors for parameter estimates when using the em algorithm several examples are discussed involving simple variance component and longitudinal models and small area estimation

generalizing logistic regression by nonparametric mixing abstract logistic regression is a common technique for analyzing the effect of a covariate vector x on the number of successes y in m trials when y has a binomial distribution but at times either the logistic curve does not describe the probability of success p x adequately or m is larger than 1 and y is more variable than the binomial distribution allows overdispersion relative to the binomial distribution is possible if the m trials in a set or litter are positively correlated an important covariate is omitted or x is measured with error a simple way to accommodate departures from the logit link and overdispersion is to introduce a random intercept and thus permit a random propensity toward success when varies between individual binary trials according to a discrete or multimodal distribution p x has smooth steps and y has a binomial m p x distribution when the random is constant for a set of m binary trials and varies between sets of m trials according to a discrete or multimodal

a diagnostic test for the mixing distribution in a generalised linear mixed model summary we introduce a diagnostic test for the mixing distribution in a generalised linear mixed model the test is based on the difference between the marginal maximum likelihood and conditional maximum likelihood estimators of a subset of the fixed effects in the model we derive the asymptotic variance of this difference and propose a test statistic that has a limiting chi squared distribution under the null hypothesis that the mixing distribution is correctly specified this strategy uses an idea presented by hausman 1978 who considered analogous tests for the linear mixed model an important advantage of the methods outlined here is that the resulting diagnostic test is easily implemented in commercial software we illustrate the method by applying it to data from a clinical trial investigating the effect of hormonal contraceptives in women

a new method for chebyshev approximation of complex valued functions problems of this type appear in many connections and have been treated by various authors cf 2 12 13 considering the computation of best rational chebyshev approximations to complex valued functions by descent algorithms ellacott and williams 6 report that the main portion of computer time is spent in the solution of the linear subproblems they have applied the very slowly convergent lawson algorithm to these subproblems and they remark that a fast linear algorithm would bring about a significant improvement in efficiency in our paper we describe a new algorithm which seems to be the first one which works for the continuous case where t is not a finite set too in our method we transform the complex approximation problem into a real linear optimization problem with infinitely many constraints a so called semi infinite program sip cf 5 8 10 this is solved by a two step procedure in the first step we apply the stabilized simplex algorithm to a discrete approximation of the sip and in the second we make use of the newton raphson method in order to obtain very rapidly the solution of the sip and thus of our original approximation problem

deconvolution methods for non parametric inference in two level mixed models summary we develop a general non parametric approach to the analysis of clustered data via random effects assuming only that the link function is known the regression functions and the distributions of both cluster means and observation errors are treated non parametrically our argument proceeds by viewing the observation error at the cluster mean level as though it were a measurement error in an errors in variables problem and using a deconvolution argument to access the distribution of the cluster mean a fourier deconvolution approach could be used if the distribution of the error in variables were known in practice it is unknown of course but it can be estimated from repeated measurements and in this way deconvolution can be achieved in an approximate sense this argument might be interpreted as implying that large numbers of replicates are necessary for each cluster mean distribution but that is not so we avoid this requirement by incorporating statistical smoothing over values of nearby explanatory variables empirical rules are developed for the choice of smoothing parameter numerical simulations and an application to real data demonstrate small sample performance for this package of methodology we also develop theory establishing statistical consistency

on the klein gordon equation and hyperbolic pseudoanalytic function theory elliptic pseudoanalytic function theory was considered independently by bers and vekua decades ago in this paper we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers we consider the klein gordon equation with a potential with the aid of one particular solution we factorize the klein gordon operator in terms of two vekua type operators we show that real parts of the solutions of one of these vekua type operators are solutions of the considered klein gordon equation using hyperbolic pseudoanalytic function theory we then obtain explicit construction of infinite systems of solutions of the klein gordon equation with potential finally we give some examples of the application of the proposed procedure

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

nonparametric estimation in nonlinear mixed effects models summary a nonparametric approach is developed herein to estimate parameters in nonlinear mixed effects models asymptotic properties of the nonparametric maximum likelihood estimators and associated computational algorithms are provided empirical bayes estimators of functionals of the random effects are also developed applications to population pharmacokinetics are given

stability and periodicity in coupled pinney equations abstract we show that the coupled pinney equations with periodic coefficient constitute a class of nonlinear coupled hill s equations for which the problems of stability and periodicity of the solutions reduce to those for a pair of uncoupled linear hill s equations one of which is of ince type

bayesian identification selection and estimation of semiparametric functions in high dimensional additive models in this paper we propose an approach to both estimate and select unknown smooth functions in an additive model with potentially many functions each function is written as a linear combination of basis terms with coefficients regularized by a proper linearly constrained gaussian prior given any potentially rank deficient prior precision matrix we show how to derive linear constraints so that the corresponding effect is identified in the additive model this allows for the use of a wide range of bases and precision matrices in priors for regularization by introducing indicator variables each constrained gaussian prior is augmented with a point mass at zero thus allowing for function selection posterior inference is calculated using markov chain monte carlo and the smoothness in the functions is both the result of shrinkage through the constrained gaussian prior and model averaging we show how using non degenerate priors on the shrinkage parameters enables the application of substantially more computationally efficient sampling schemes than would otherwise be the case we show the favourable performance of our approach when compared to two contemporary alternative bayesian methods to highlight the potential of our approach in high dimensional settings we apply it to estimate two large seemingly unrelated regression models for intra day electricity load both models feature a variety of different univariate and bivariate functions which require different levels of smoothing and where component selection is meaningful priors for the error disturbance covariances are selected carefully and the empirical results provide a substantive contribution to the electricity load modelling literature in their own right

representation of solutions to the one dimensional schrodinger equation in terms of neumann series of bessel functions abstract a new representation of solutions to the equation y q x y 2 y is obtained for every x the solution is represented as a neumann series of bessel functions depending on the spectral parameter due to the fact that the representation is obtained using the corresponding transmutation operator a partial sum of the series approximates the solution uniformly with respect to which makes it especially convenient for the approximate solution of spectral problems the numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy

smooth random effects distribution in a linear mixed model a linear mixed model with a smooth random effects density is proposed a similar approach to p spline smoothing of eilers and marx 1996 statistical science 11 89 121 is applied to yield a more flexible estimate of the random effects density our approach differs from theirs in that the b spline basis functions are replaced by approximating gaussian densities fitting the model involves maximizing a penalized marginal likelihood the best penalty parameters minimize akaike s information criterion employing gray s 1992 journal of the american statistical association 87 942 951 results although our method is applicable to any dimensions of the random effects structure in this article the two dimensional case is explored our methodology is conceptually simple and it is relatively easy to fit in practice and is applied to the cholesterol data first analyzed by zhang and davidian 2001 biometrics 57 795 802 a simulation study shows that our approach yields almost unbiased estimates of the regression and the smoothing parameters in small sample settings consistency of the estimates is shown in a particular case

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

diagnosis of random effect model misspecification in generalized linear mixed models for binary response summary generalized linear mixed models glmms are widely used in the analysis of clustered data however the validity of likelihood based inference in such analyses can be greatly affected by the assumed model for the random effects we propose a diagnostic method for random effect model misspecification in glmms for clustered binary response we provide a theoretical justification of the proposed method and investigate its finite sample performance via simulation the proposed method is applied to data from a longitudinal respiratory infection study

linear regression with interval censored data i occurred in reliability studies destructive tests are often used to find whether an item e g fire extinguisher has failed in rodent bioassay experiments the presence or absence of a tumor may only be detected through sacrifices for certain designs of experiments t are i i d random variables with a i known probability density in general t and z are associated in case 2 ii interval censoring two censoring variables t t are observed with the ii knowledge that y is inside the interval t t in both cases of interval ii i

large scale multiple testing under dependence summary the paper considers the problem of multiple testing under dependence in a compound decision theoretic framework the observed data are assumed to be generated from an underlying two state hidden markov model we propose oracle and asymptotically optimal datadriven procedures that aim to minimize the false non discovery rate fnr subject to a constraint on the false discovery rate fdr it is shown that the performance of a multiple testing procedure can be substantially improved by adaptively exploiting the dependence structure among hypotheses and hence conventional fdr procedures that ignore this structural information are inefficient both theoretical properties and numerical performances of the procedures proposed are investigated it is shown that the procedures proposed control fdr at the desired level enjoy certain optimality properties and are especially powerful in identifying clustered non null cases the new procedure is applied to an influenza like illness surveillance study for detecting the timing of epidemic periods

oracle and adaptive compound decision rules for false discovery rate control we develop a compound decision theory framework for multiple testing problems and derive an oracle rule based on the z values that minimizes the false nondiscovery rate fnr subject to a constraint on the false discovery rate fdr we show that many commonly used multiple testing procedures which are p value based are inefficient and propose an adaptive procedure based on the z values the z value based adaptive procedure asymptotically attains the performance of the z value oracle procedure and is more efficient than the conventional p value based methods we investigate the numerical performance of the adaptive procedure using both simulated and real data in particular we demonstrate our method in an analysis of the microarray data from a human immunodeficiency virus study that involves testing a large number of hypotheses simultaneously

modified spectral parameter power series representations for solutions of sturm liouville equations and their applications abstract spectral parameter power series spps representations for solutions of sturm liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems they are based on a computation of recursive integrals sometimes called formal powers in this paper new relations between the formal powers are presented which considerably improve and extend the application of the spps method for example originally the spps method at a first step required to construct a nonvanishing in general a complex valued particular solution corresponding to the zero value of the spectral parameter the obtained relations remove this limitation additionally equations with nasty sturm liouville coefficients 1 p or r can be solved by the spps method we develop the spps representations for solutions of sturm liouville equations of the form p x u q x u k 1 n k r k u x a b where r k u r k x u s k x u k 1 n the complex valued functions p q r k s k are continuous on the finite segment a b several numerical examples illustrate the efficiency of the method and its wide applicability

a semiparametric mixture approach to case control studies with errors in covariables abstract methods are devised for estimating the parameters of a prospective logistic model in a case control study with dichotomous response d that depends on a covariate x for a portion of the sample both the gold standard x and a surrogate covariate w are available however for the greater portion of the data only the surrogate covariate w is available by using a mixture model the relationship between the true covariate and the response can be modeled appropriately for both types of data the likelihood depends on the marginal distribution of x and the measurement error density w x d the latter is modeled parametrically based on the validation sample the marginal distribution of the true covariate is modeled using a nonparametric mixture distribution in this way we can improve the efficiency and reduce the bias of the parameter estimates the results also apply when there is no validation data provided the error distribution is known or estimated from an independent data source many of the

analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems a method for approximate solution of spectral problems for sturm liouville equations based on the construction of the delsarte transmutation operators is presented in fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in khmelnytskaya et al 2013 and kravchenko and torba 2014 the method allows one to compute both lower and higher eigendata with an extreme accuracy

on 2 1 dimensional ermakov systems ermakov type systems in 2 1 dimensions are introduced multiwave solutions of a 2 1 dimensional pinney equation and a modulated 2 1 dimensional sine gordon equation are thereby constructed

rational ermakov systems of fuchsian type studies a subclass of a class of systems of coupled nonlinear oscillators that have well behaved singularities which can be studied using an exact linearization this allows the author to classify such systems and to isolate those with special properties

standard transmutation operators for the one dimensional schrodinger operator with a locally integrable potential abstract we study a special class of operators t satisfying the transmutation relation d 2 d x 2 q t u t d 2 d x 2 u in the sense of distributions where q is a locally integrable function and u belongs to a suitable space of distributions depending on the smoothness properties of q a method which allows one to construct a fundamental set of transmutation operators of this class in terms of a single particular transmutation operator is presented moreover following 27 we show that a particular transmutation operator can be represented as a volterra integral operator of the second kind we study the boundedness and invertibility properties of the transmutation operators and use these to obtain a representation for the general distributional solution of the equation d 2 u d x 2 q u u c in terms of the general solution of the same equation with 0

estimation of integrated squared density derivatives from a contaminated sample we propose a kernel estimator of integrated squared density derivatives from a sample that has been contaminated by random noise we derive asymptotic expressions for the bias and the variance of the estimator and show that the squared bias term dominates the variance term this coincides with results that are available for non contaminated observations we then discuss the selection of the bandwidth parameter when estimating integrated squared density derivatives based on contaminated data we propose a data driven bandwidth selection procedure of the plug in type and investigate its finite sample performance via a simulation study

a smooth nonparametric estimate of a mixing distribution using mixtures of gaussians abstract we propose a method of estimating mixing distributions using maximum likelihood over the class of arbitrary mixtures of gaussians subject to the constraint that the component variances be greater than or equal to some minimum value h this approach can lead to estimates of many shapes with smoothness controlled by parameter h we show that the resulting estimate will always be a finite mixture of gaussians each having variance h the nonparametric maximum likelihood estimate can be viewed as a special case with h 0 the method can be extended to estimate multivariate mixing distributions examples and the results of a simulation study are presented

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

prediction of random effects in linear and generalized linear models under model misspecification statistical models that include random effects are commonly used to analyze longitudinal and correlated data often with the assumption that the random effects follow a gaussian distribution via theoretical and numerical calculations and simulation we investigate the impact of misspecification of this distribution on both how well the predicted values recover the true underlying distribution and the accuracy of prediction of the realized values of the random effects we show that although the predicted values can vary with the assumed distribution the prediction accuracy as measured by mean square error is little affected for mild to moderate violations of the assumptions thus standard approaches readily available in statistical software will often suffice the results are illustrated using data from the heart and estrogen progestin replacement study using models to predict future blood pressure values

sieve maximum likelihood estimator for semiparametric regression models with current status data in a randomized controlled clinical trial study where the response variable of interest is the time to occurrence of a certain event it is often too expensive or even impossible to observe the exact time however the current status of the subject at a random time of inspection is much more natural feasible and practical in terms of cost effectiveness this article considers a semiparametric regression model that consists of parametric and nonparametric regression components a sieve maximum likelihood estimator mle is proposed to estimate the regression parameter allowing exploration of the nonlinear relationship between a certain covariate and the response function asymptotic properties of the proposed sieve mles are discussed under some mild conditions the estimators are shown to be strongly consistent moreover the estimators of the unknown parameters are asymptotically efficient and normally distributed and the estimator of the nonparametric function has an optimal convergence rate simula

wave polynomials transmutations and cauchy s problem for the klein gordon equation abstract we prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials solutions of the klein gordon equation with a variable coefficient using the transmutation transformation operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the cauchy problem for the klein gordon equation based on this result we develop a numerical method for solving the cauchy problem and test its performance

semi nonparametric maximum likelihood estimation on developpe une approche basee sur une innovation due a phillips 1983 sur les approximations des fonctions de densite

innovations in bayes and empirical bayes methods estimating parameters populations and ranks by formalizing the relation among components and borrowing information among them bayes and empirical bayes methods can produce more valid efficient and informative statistical evaluations than those based on traditional methods in addition bayesian structuring of complicated models and goals guides development of appropriate statistical approaches and generates summaries which properly account for sampling and modelling uncertainty computing innovations enable implementation of complex and relevant models thereby substantially increasing the role of bayes empirical bayes methods in important statistical assessments policy relevant statistical assessments involve synthesis of information from a set of related components such as medical clinics geographic regions or research studies typical assessments include inference for individual parameters synthesis over the collection of components for example the parameter histogram and comparisons among parameters for example ranks the relative importance of these goals depends on the context bayesian structuring provides a guide to valid inference for example while posterior means are the obvious and optimal estimates for individual components under squared error loss their empirical distribution function edf is underdispersed and never valid for estimating the edf of the true underlying parameters effective histogram estimates result from optimizing a loss function based in a distance between the histogram and its estimate similarly ranking observed data usually produces poor estimates and ranking posterior means can be inappropriate effective estimates should be based on a loss function that caters directly to ranks using examples of borrowing information shrinkage and the variance bias trade off we motivate bayes and empirical bayes analysis then we outline the formal approach and discuss triple goal estimates with values that when ranked produce optimal ranks for which the edf is an optimal estimate of the parameter edf and such that the values themselves are effective estimates of co ordinate specific parameters we use basic models and data analysis examples to highlight the conceptual and structural issues

newton raphson and em algorithms for linear mixed effects models for repeated measures data abstract we develop an efficient and effective implementation of the newton raphson nr algorithm for estimating the parameters in mixed effects models for repeated measures data we formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by jennrich and schluchter 1986 to speed convergence and ensure a positive definite covariance matrix for the random effects at each iteration we use matrix decompositions to develop efficient and computationally stable implementations of both the nr algorithm and an em algorithm laird and ware 1982 for this model we compare the two methods em vs nr in terms of computational order and performance on two sample data sets and conclude that in most situations a well implemented nr algorithm is preferable to the em algorithm or em algorithm with aitken s acceleration the term repeated measures refers to experimental designs where there are several individuals and several

mixed effect statistics for group analysis in fmri a nonparametric maximum likelihood approach this technical note describes a collection of test statistics accounting for estimation uncertainties at the within subject level that can be used as alternatives to the standard t statistic in one sample random effect analyses i e when testing the mean effect of a population we build such test statistics by estimating the across subject distribution of the effects using maximum likelihood under a nonparametric mixed effect model for inference purposes the statistics are calibrated using permutation tests to achieve exact false positive control under a symmetry assumption regarding the across subject distribution the new tests are implemented in a freely available toolbox for spm called distance

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

on the linearization of the generalized ermakov systems a linearization procedure is proposed for ermakov systems with frequency depending on dynamic variables the procedure applies to a wide class of generalized ermakov systems which are shown to be linearizable in a manner similar to that applicable to usual ermakov systems the kepler ermakov systems belong to this category but other more generic systems are also included