dose response signal detection under model uncertainty we investigate likelihood ratio contrast tests for dose response signal detection under model uncertainty when several competing regression models are available to describe the dose response relationship the proposed approach uses the complete structure of the regression models but does not require knowledge of the parameters of the competing models standard likelihood ratio test theory is applicable in linear models as well as in nonlinear regression models with identifiable parameters however for many commonly used nonlinear dose response models the regression parameters are not identifiable under the null hypothesis of no dose response and standard arguments cannot be used to obtain critical values we thus derive the asymptotic distribution of likelihood ratio contrast tests in regression models with a lack of identifiability and use this result to simulate the quantiles based on gaussian processes the new method is illustrated with a real data example and compared to existing procedures using theoretical investigations as well as simulations

asymptotic formulae for the quantum renyi entropies of position application to the infinite well general asymptotic formulae are derived by means of the wkb approximation for the continuous and discrete renyi entropies of position of one dimensional quantum systems in energy eigenstates in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems these results are checked in the simplest particular case of the infinite potential well where the asymptotic formula for continuous entropies holds as an exact identity for the discrete entropies analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified

splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

standard and entropic uncertainty relations of the finite well this paper deals with the standard and entropic uncertainty relations for the finite potential well with different length and potential depth we calculate the position and momentum dispersions and entropies that are dependent on the quantum states of the finite well and determine the corresponding standard and entropic uncertainty relations then we calculate the entropy of occurrence for the mass point inside and outside the well we discuss the results and determine a class of probability distributions with a parameter for which the standard deviation becomes infinite independent of the value of this parameter

bayesian isotonic regression and trend analysis in many applications the mean of a response variable can be assumed to be a nondecreasing function of a continuous predictor controlling for covariates in such cases interest often focuses on estimating the regression function while also assessing evidence of an association this article proposes a new framework for bayesian isotonic regression and order restricted inference approximating the regression function with a high dimensional piecewise linear model the nondecreasing constraint is incorporated through a prior distribution for the slopes consisting of a product mixture of point masses accounting for flat regions and truncated normal densities to borrow information across the intervals and smooth the curve the prior is formulated as a latent autoregressive normal process this structure facilitates efficient posterior computation since the full conditional distributions of the parameters have simple conjugate forms point and interval estimates of the regression function and posterior probabilities of an association for different regions of the predictor can be estimated from a single mcmc run generalizations to categorical outcomes and multiple predictors are described and the approach is applied to an epidemiology application

feasible nonparametric estimation of multiargument monotone functions abstract this article presents a two stage estimation procedure that uses an ad hoc but very easily implemented isotonization of a kernel estimator this procedure yields an isotonic estimator with the convergence properties of the kernel estimator although the isotonization in the second stage does not satisfy the least squares condition this hybrid estimator may be considered to be a multidimensional generalization of similar procedures for the one dimensional case suggested by friedman and tibshirani and by mukarjee we derive some of the asymptotic properties of our estimator and demonstrate other statistical properties with monte carlo studies we conclude by providing a real data example

bayesian nonparametric inference for quantal response data the problem of nonparametric bayes estimation of a tolerance distribution based on quantal response data has been considered previously with a prior distribution based on the dirichlet process in the present article a broad class of priors is developed for this problem by allowing the hazard function of the tolerance distribution to be a realization of a nonnegative stochastic process with independent increments this class includes the dirichlet prior as a special case in addition priors over a space of absolutely continuous tolerance distributions which includes ifr dfr and u shaped failure rate distributions are constructed by taking the failure rate to be the superposition of two processes with independent increments posterior laplace transforms of the corresponding processes are obtained based on quantal response data with binomial sampling these posterior laplace transforms are then used to find bayes estimates and examples are given to illustrate the results

information entropy of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy e n of the classical orthogonal polynomials e email protected p 2 n x logp 2 n x w x dx here an explicit formula for the entropy of general orthogonal polynomials is obtained in terms of the eigenvalues of jacobi matrix associated with the orthogonal polynomials p n x and the weighted moments of p 2 n x m n email protected x kp 2 n x w x dx k 1 2

testing and estimating shape constrained nonparametric density and regression in the presence of measurement error in many applications we can expect that or are interested to know if a density function or a regression curve satisfies some specific shape constraints for example when the explanatory variable x represents the value taken by a treatment or dosage the conditional mean of the response y is often anticipated to be a monotone function of x indeed if this regression mean is not monotone in the appropriate direction then the medical or commercial value of the treatment is likely to be significantly curtailed at least for values of x that lie beyond the point at which monotonicity fails in the case of a density common shape constraints include log concavity and unimodality if we can correctly guess the shape of a curve then nonparametric estimators can be improved by taking this information into account addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint and if the conclusion of the test is positive a technique for e

unobserved heterogeneity in auctions under restricted stochastic dominance abstract we study the identification of first price auctions with nonseparable unobserved heterogeneity in particular we extend hu et al 2013 by relaxing the first order stochastic dominance condition instead we assume restricted stochastic dominance relations among value quantile functions and show that the same relations pass to bid quantile functions an ordered tree summarizes these relations and provides a total ordering relying on the proposed restricted stochastic dominance ordering we extend a list of identification results in the empirical auction literature

mass recentred kernel smoothers the local linear smoother usually gives better performance than the nadaraya watson smoother an exception is the case of data sparsity here we discuss a modification of the nadaraya watson smoother by muller song 1993 based on a horizontal shift of the kernel weights towards the local centre of mass of the design points this gives performance similar to the local linear when that works well and better performance when it does not the new smoother also preserves monotonicity shifting towards the centre of mass is also used to develop a modified kernel density estimate which cancels the well known peak spreading effect

entropic integrals of hyperspherical harmonics and spatial entropy of d dimensional central potentials the information entropy of a single particle in a quantum mechanical d dimensional central potential is separated in two parts one depends only on the specific form of the potential radial entropy and the other depends on the angular distribution spatial entropy the latter is given by an entropic like integral of the hyperspherical harmonics which is expressed in terms of the entropy of the gegenbauer polynomials this entropy is expressed in terms of the values of the quadratic logarithmic potential of gegenbauer polynomials cn t at the zeros of these polynomials then this potential for integer is given as a finite expansion of chebyshev polynomials of even order whose coefficients are shown to be wilson polynomials

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

bayesian local extremum splines we consider shape restricted nonparametric regression on a closed set formula see text where it is reasonable to assume the function has no more than h local extrema interior to formula see text following a bayesian approach we develop a nonparametric prior over a novel class of local extremum splines this approach is shown to be consistent when modeling any continuously differentiable function within the class considered and is used to develop methods for testing hypotheses on the shape of the curve sampling algorithms are developed and the method is applied in simulation studies and data examples where the shape of the curve is of interest

estimating smooth monotone functions many situations call for a smooth strictly monotone function f of arbitrary flexibility the family of functions defined by the differential equation d 2 f w df where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions the solution to this equation is f c 0 c 1 d 1 exp d 1 w where c 0 and c 1 are arbitrary constants and d 1 is the partial integration operator a basis for expanding w is suggested that permits explicit integration in the expression of f in fitting data it is also useful to regularize f by penalizing the integral of w 2 since this is a measure of the relative curvature in f applications are discussed to monotone nonparametric regression to the transformation of the dependent variable in non linear regression and to density estimation

illustration of bayesian inference in normal data models using gibbs sampling abstract the use of the gibbs sampler as a method for calculating bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models including variance components unordered and ordered means hierarchical growth curves and missing data in a crossover trial in all cases the approach is straightforward to specify distributionally and to implement computationally with output readily adapted for required inference summaries

bounding quantile demand functions using revealed preference inequalities this paper develops a new technique for the estimation of consumer demand models with unobserved heterogeneity subject to revealed preference inequality restrictions particular attention is given to nonseparable heterogeneity the inequality restrictions are used to identify bounds on quantile demand functions a nonparametric estimator for these bounds is developed and asymptotic properties are derived an empirical application using data from the u k family expenditure survey illustrates the usefulness of the methods by deriving bounds and con dence sets for estimated quantile demand functions

nonparametric bayesian bioassay with prior constraints on the shape of the potency curve summary we consider the problem of implementing nonparametric bayesian analysis of bioassay problems when the prior constrains the form of the potency curve for example to be convex concave or ogive sampling based approaches to calculating posterior features of interest are detailed and illustrated extensions to prior constraints on hazard rates are outlined

a nonparametric bayesian modeling approach for cytogenetic dosimetry summary in cytogenetic dosimetry samples of cell cultures are exposed to a range of doses of a given agent in each sample at each dose level some measure of cell disability is recorded the objective is to develop models that explain cell response to dose such models can be used to predict response at unobserved doses more important such models can provide inference for unknown exposure doses given the observed responses typically cell disability is viewed as a poisson count but in the present work a more appropriate response is a categorical classification in the literature modeling in this case is very limited what exists is purely parametric we propose a fully bayesian nonparametric approach to this problem we offer comparison with a parametric model through a simulation study and the analysis of a real dataset modeling blood cultures exposed to radiation where classification is with regard to number of micronuclei per cell

nonparametric instrumental variable estimation under monotonicity the ill posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suer from a very slow logarithmic rate of convergence in this paper we show that restricting the problem to models with monotone regression functions and monotone instruments signicantly weakens the ill posedness of the problem in stark contrast to the existing literature the presence of a monotone instrument implies boundedness of our measure of ill posedness when restricted to the space of monotone functions based on this result we derive a novel non asymptotic error bound for the constrained estimator that imposes monotonicity of the regression function for a given sample size the bound is independent of the degree of ill posedness as long as the regression function is not too steep as an implication the bound allows us to show that the constrained estimator converges at a fast polynomial rate independently of the degree of ill posedness in a large but slowly shrinking neighborhood of constant functions our simulation study demonstrates signicant nite sample performance gains from imposing monotonicity even when the regression function is rather far from being a constant we apply the constrained estimator to the problem of estimating gasoline demand functions from u s data

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

monotonic smoothing splines fitted by cross validation a practical method for calculating monotonic cubic smoothing splines is given linear sufficient conditions for monotonicity are employed and the spline coefficients are obtained using quadratic p

a class of grouped brunk estimators and penalized spline estimators for monotone regression we study a class of monotone univariate regression estimators we use b splines to approximate an underlying regression function and estimate spline coefficients based on grouped data we investigate asymptotic properties of two monotone estimators a grouped brunk estimator and a penalized monotone estimator these estimators are consistent at the boundary and their mean square errors achieve optimal convergence rates under suitable assumptions of the true regression function asymptotic distributions are developed and are shown to be independent of spline degrees and the number of knots simulation results and car data illustrate performance of the proposed estimators

wavelets and orthogonal polynomials based on harmonic oscillator eigenstates we present a new method for the construction of wavelets the main tool is the harmonic oscillator eigenstates with the help of their properties we are able to construct wavelets which have explicit expressions and have exponential decay in both time domain and frequency domain moreover we get a new family of polynomials which are orthogonal in a weighted squarely integrable function space

bayesian isotonic density regression density regression models allow the conditional distribution of the response given predictors to change flexibly over the predictor space such models are much more flexible than nonparametric mean regression models with nonparametric residual distributions and are well supported in many applications a rich variety of bayesian methods have been proposed for density regression but it is not clear whether such priors have full support so that any true data generating model can be accurately approximated this article develops a new class of density regression models that incorporate stochastic ordering constraints which are natural when a response tends to increase or decrease monotonely with a predictor theory is developed showing large support methods are developed for hypothesis testing with posterior computation relying on a simple gibbs sampler frequentist properties are illustrated in a simulation study and an epidemiology application is considered

linear bayes estimators of the potency curve in bioassay abstract the bayesian nonparametric approach to estimating the tolerance distribution in quantal bioassay has received some attention the computational difficulty in evaluating these bayes estimators has hindered their applications this paper explores the linear bayes approach to the bioassay problem these linear bayes estimators can be computed easily by using statistical software which has the capability of inverting a matrix let us state the quantal bioassay problem as follows the experimenter intends to test the potency of a stimulus by giving subjects injections of the stimulus at different levels namely he chooses l dosage levels t sub 1 t sub l and treats n sub 1 n sub l subjects at these levels respectively each subject possesses a fixed tolerance level if a stimulus exceeds a subject s tolerance level the subject responds positively if not there is no response therefore we observe the number of positive responses at each level these numbers are denoted by k sub 1 k sub l the potency curve f is the distribution of tolerance levels i e f is defined by the probability f t of getting a positive response to a dosage at level t for all t the objective of this article is to make inferences about the potency curve f keywords ferguson s dirichlet process

bayesian nonparametric estimation of continuous monotone functions with applications to dose response analysis in this article we consider monotone nonparametric regression in a bayesian framework the monotone function is modeled as a mixture of shifted and scaled parametric probability distribution functions and a general random probability measure is assumed as the prior for the mixing distribution we investigate the choice of the underlying parametric distribution function and find that the two sided power distribution function is well suited both from a computational and mathematical point of view the model is motivated by traditional nonlinear models for dose response analysis and provides possibilities to elicitate informative prior distributions on different aspects of the curve the method is compared with other recent approaches to monotone nonparametric regression in a simulation study and is illustrated on a data set from dose response analysis

glassy properties of frustrated arrays of nonlinear devices we study linear arrays of different number of quartic oscillators shaped in the form of a ring when gaussian noise temperature is added frustration is introduced through periodic boundary conditions and repulsive directional interactions between neighboring oscillators we show that these systems have similar dynamic properties than the arrays of fluxgates magnetometers we find that there is a critical number of oscillators separating the regimes arising for systems with few and many oscillators and show that they reach an optimum ordering for a nonvanishing temperature we also find that they have a relaxation process with an infinite mean life that is typical of glassy systems

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

nonparametric kernel regression subject to monotonicity constraints we suggest a method for monotonizing general kernel type estimators for example local linear estimators and nadaraya watson estimators attributes of our approach include the fact that it produces smooth estimates indeed with the same smoothness as the unconstrained estimate the method is applicable to a particularly wide range of estimator types it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented therefore an experimenter may use his or her favorite kernel estimator and their favorite bandwidth selector to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way implementation involves only an off the shelf programming routine the method is based on maximizing fidelity to the conventional empirical approach subject to monotonicity we adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change in the sense of a distance measure subject to imposing the constraint of monotonicity

the reduced monotonic regression method abstract medical researchers often desire to categorize patients into monotonic response groups based on the relationship between continuous variables isotonic regression fits consist of level sets of increasing value for which the estimated response is constant however the number of level sets obtained is often large preventing simple description this article introduces two new nonparametric methods called reduced isotonic regression and reduced monotonic regression the latter being a two sided extension of the former for use when the direction of the trend is unknown using a backward elimination algorithm the new procedures reduce the number of level sets by combining those whose values do not differ greatly for the statistical relations examined here the reduced monotonic method averaged at most 30 of the number of level sets obtained for isotonic regression the method is illustrated with an example that examines the relationship between risk factors for survival among children with leukem

bayesian nonparametric inference on the dose level with specified response rate the richness of nonparametric bayesian models has attracted many different applications its application in dose finding studies has been hindered due to lack of methodologies on the nonparametric bayesian inference on percentiles the primary interest in dose finding studies focuses inference on the unknown toxicity or efficacy dose level corresponding to a prespecified rate this paper shows how this problem may generally be handled by deriving inference on percentiles of a distribution following a dirichlet process prior in particular theoretical results are derived to obtain the nonparametric bayesian inference of the unknown dose level this is followed by a description of the numerical implementation of that theory the method also allows efficient estimation of the entire potency curve finally the usefulness of the approach is demonstrated via an experimental data example

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

an approximate dynamic programming algorithm for monotone value functions many sequential decision problems can be formulated as markov decision processes mdps where the optimal value function or cost to go function can be shown to satisfy a monotone structure in some or all of its dimensions when the state space becomes large traditional techniques such as the backward dynamic programming algorithm i e backward induction or value iteration may no longer be effective in finding a solution within a reasonable time frame and thus we are forced to consider other approaches such as approximate dynamic programming adp we propose a provably convergent adp algorithm called monotone adp that exploits the monotonicity of the value functions to increase the rate of convergence in this paper we describe a general finite horizon problem setting where the optimal value function is monotone present a convergence proof for monotone adp under various technical assumptions and show numerical results for three application domains optimal stopping energy storage allocation and glycemic control for diabetes patients the empirical results indicate that by taking advantage of monotonicity we can attain high quality solutions within a relatively small number of iterations using up to two orders of magnitude less computation than is needed to compute the optimal solution exactly

posterior distribution of a dirichlet process from quantal response data the posterior distribution of a dirichlet process from n quantal responses at r dosage levels has been recognized by antoniak as a mixture of dirichlet processes the purpose of this paper is to develop a systematic procedure for computing finite dimensional distributions of such mixtures which can be equivalently expressed as multivariate beta distributions with random parameter vectors it is shown that the sequence of random parameter vectors of the updated beta posteriors from observations at increasing dosage levels evolves in a manner which is described by r separate markov chains this description is then used to derive the asymptotic posterior distribution the weak limits of the relevant markov chains are shown to be solutions of certain stochastic differential equations and the random parameter vector of the posterior beta distribution is shown to be asymptotically normal the mean vector and covariance matrix of which are given by recursion formulas

bayesian nonparametric modeling using mixtures of triangular distributions nonparametric modeling is an indispensable tool in many applications and its formulation in an hierarchical bayesian context using the entire posterior distribution rather than particular expectations increases its flexibility in this article the focus is on nonparametric estimation through a mixture of triangular distributions the optimality of this methodology is addressed and bounds on the accuracy of this approximation are derived although our approach is more widely applicable we focus for simplicity on estimation of a monotone nondecreasing regression on 0 1 with additive error effectively approximating the function of interest by a function having a piecewise linear derivative computationally accessible methods of estimation are described through an amalgamation of existing markov chain monte carlo algorithms simulations and examples illustrate the approach

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

logarithmic potential of hermite polynomials and information entropies of the harmonic oscillator eigenstates the problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential vn t hn x 2 ln x t e x2 dx at the zeros of the hermite polynomial hn x here a closed analytical expression for vn t is obtained which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of hn x is known an inequality for the values of vn t at the zeros of hn x is conjectured which leads to a new nonvariational upper bound for the entropies finally the exact formula for vn t is written in an alternative way which allows the entropies to be expressed in terms of the even order spectral moments of the hermite polynomials the asymptotic n 1 limit of this alternative expression for the entropies is discussed and the conjectured upper bound for the entropies is proved to be asymptotically valid

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

monotone b spline smoothing abstract estimation of growth curves or item response curves often involves monotone data smoothing methods that have been studied in the literature tend to be either less flexible or more difficult to compute when constraints such as monotonicity are incorporated built on the ideas of koenker ng and portnoy and ramsay we propose monotone b spline smoothing based on l 1 optimization this method inherits the desirable properties of spline approximations and the computational efficiency of linear programs the constrained fit is similar to the unconstrained estimate in terms of computational complexity and asymptotic rate of convergence through applications to some real and simulated data we show that the method is useful in a variety of applications the basic ideas utilized in monotone smoothing can be useful in some other constrained function estimation problems

entropy of orthogonal polynomials with freud weights and information entropies of the harmonic oscillator potential the information entropy of the harmonic oscillator potential v x 1 2 x2 in both position and momentum spaces can be expressed in terms of the so called entropy of hermite polynomials i e the quantity sn h h2n x log h2n x e x2dx these polynomials are instances of the polynomials orthogonal with respect to the freud weights w x exp x m m 0 here a very precise and general result of the entropy of freud polynomials recently established by aptekarev et al j math phys 35 4423 4428 1994 specialized to the hermite kernel case m 2 leads to an important refined asymptotic expression for the information entropies of very excited states i e for large n in both position and momentum spaces to be denoted by s and s respectively briefly it is shown that for large values of n s 1 2log log 2n e o 1 and s 1 2log log 2n e o 1 so that s s log 2 2n e2 o 1 in agreement with the generalized indetermination relation of byalinicki birula and mycielski commun m

comparing the shapes of regression functions summary does a regression function follow a specified shape do two regression functions have the same shape how can regression functions be grouped based on shape these questions can occur when investigating monotonicity when counting local maxima or when studying variation in families of curves one can address these questions by considering the rank correlation coefficient between two functions this correlation is a generalisation of the rank correlation between two finite sets of numbers and is equal to one if and only if the two functions have the same shape a sample rank correlation based on smoothed estimates of the regression functions consistently estimates the true correlation this sample rank correlation can be used as a measure of similarity between functions in cluster analysis and as a measure of monotonicity or modality

nonparametric option pricing under shape restrictions frequently economic theory places shape restrictions on functional relationships between economic variables this paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators we apply this technique to estimate the state price density spd or risk neutral density implicit in the market prices of options the option pricing function must be monotonic and convex simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day to day option pricing once appropriate theory motivated shape restrictions are imposed using s p 500 option prices we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999 unlike the constrained estimator we propose

bayesian nonparametric inference for effective doses in a quantal response experiment ramsey 1972 biometrics 28 841 858 developed a prior distribution for a single response curve arising from an indirect biological assay based upon quantal responses this paper presents the derivation of the exact prior and posterior distributions of an effective dose which follow from ramsey s prior since the posterior distribution of an effective dose is numerically difficult to compute in a large experiment methods for approximating this distribution are given numerical examples are presented to show the accuracy of the approximate methods finally the usefulness of the present results in assessing ramsey s prior is illustrated

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

variable selection in monotone single index models via the adaptive lasso we consider the problem of variable selection for monotone single index models a single index model assumes that the expectation of the outcome is an unknown function of a linear combination of covariates assuming monotonicity of the unknown function is often reasonable and allows for more straightforward inference we present an adaptive lasso penalized least squares approach to estimating the index parameter and the unknown function in these models for continuous outcome monotone function estimates are achieved using the pooled adjacent violators algorithm followed by kernel regression in the iterative estimation process a linear approximation to the unknown function is used therefore reducing the situation to that of linear regression and allowing for the use of standard lasso algorithms such as coordinate descent results of a simulation study indicate that the proposed methods perform well under a variety of circumstances and that an assumption of monotonicity when appropriate noticeably improves performance the proposed methods are applied to data from a randomized clinical trial for the treatment of a critical illness in the intensive care unit

inference for monotone functions under short and long range dependence confidence intervals and new universal limits abstractwe introduce new point wise confidence interval estimates for monotone functions observed with additive dependent noise our methodology applies to both short and long range dependence regimes for the errors the interval estimates are obtained via the method of inversion of certain discrepancy statistics this approach avoids the estimation of nuisance parameters such as the derivative of the unknown function which previous methods are forced to deal with the resulting estimates are therefore more accurate stable and widely applicable in practice under minimal assumptions on the trend and error structure the dependence of the errors especially long range dependence leads to new phenomena where new universal limits based on convex minorant functionals of drifted fractional brownian motion emerge some extensions to uniform confidence bands are also developed supplementary materials for this article are available online

a bayesian approach to the analysis of quantal bioassay studies using nonparametric mixture models we develop a bayesian nonparametric mixture modeling framework for quantal bioassay settings the approach is built upon modeling dose dependent response distributions we adopt a structured nonparametric prior mixture model which induces a monotonicity restriction for the dose response curve particular emphasis is placed on the key risk assessment goal of calibration for the dose level that corresponds to a specified response the proposed methodology yields flexible inference for the dose response relationship as well as for other inferential objectives as illustrated with two data sets from the literature

on estimation for monotone dose response curves abstract estimating monotone dose response curves with independent normal errors has been studied in detail in the literature a simple procedure using a generalization of the studentized maximum modulus technique is proposed for constructing confidence bands for monotone regression functions this procedure is shown to compare well with its predecessors the center of this smooth and efficient confidence band is then used to estimate a smooth monotone regression function its pointwise mean squared errors compare favorably with the maximum likelihood estimate a weighted average of the isotonic regression and the least squares line and the rank transformed regression a numerical example of a binding inhibition assay is included to illustrate the proposed interval and point estimation procedures

spatial entropy of central potentials and strong asymptotics of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy sn of the classical orthogonal polynomials here an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained it is shown that this entropy is intimately related to the relative entropy i 0 of the equilibrium measure 0 x and the weight function x of the polynomials to do so the theory of strong asymptotics of orthogonal polynomials on compact sets is used

nonparametric combinatorial regression for shape constrained modeling we introduce a unified approach for calculating nonparametric shape constrained regression enforcement of the shape constraint often accounts for the impact of a physical phenomenon or a specific property it also improves the model s predicability and facilitates subsequent optimizations the regression models are built by transforming the problem into the combinatorial domain where the shape constraints are imposed by bounding the combinatorial search space we start by addressing isotonicity shape constraint using a dynamic programming algorithm and demonstrate how the problem can be mapped to the graph combinatorics domain next we show how a number of other important shape constraints including unimodality convexity limited level set and limited slope can be addressed using the same framework the flexibility of proposed framework enables solving the shape constrained regression problem with an arbitrary user defined error metric this flexibility is exploited to add robustness against outliers to the model the algorithms are described in detail and their computational complexity is established the performance and effectiveness of the shape constrained regression is evaluated on traces of temperature and humidity measurements from a deployed sensor network where a high degree of accuracy and robustness is demonstrated

bayesian spatial monotonic multiple regression we consider monotonic multiple regression for contiguous regions the regression functions vary regionally and may exhibit spatial structure we develop bayesian nonparametric methodology that permits estimation of both continuous and discontinuous functional shapes using marked point process and reversible jump markov chain monte carlo techniques spatial dependence is incorporated by a flexible prior distribution which is tuned using cross validation and bayesian optimization we derive the mean and variance of the prior induced by the marked point process approach asymptotic results show consistency of the estimated functions posterior realizations enable variable selection the detection of discontinuities and prediction in simulations and in an application to a norwegian insurance data set our methodology shows better performance than existing approaches

some stochastics on monotone functions abstract a measure of departures of monotonicity of a given function the l r dip 1 r is introduced our analysis is performed to cover two different situations when the function is known our interest is related to its behavior in a stochastic model however in most cases the knowledge of the function is obtained through a preliminary estimation of the function in both situations the aim focuses in the obtainment of strong consistency results

nonparametric estimation of stage occupation probabilities in a multistage model with current status data multistage models are used to describe individuals or experimental units moving through a succession of stages corresponding to distinct states e g healthy diseased diseased with complications dead the resulting data can be considered to be a form of multivariate survival data containing information about the transition times and the stages occupied traditional survival analysis is the simplest example of a multistage model where individuals begin in an initial stage say alive and move irreversibly to a second stage death in this article we consider general multistage models with a directed tree structure progressive models in which individuals traverse through stages in a possibly non markovian manner we construct nonparametric estimators of stage occupation probabilities and marginal cumulative transition hazards empirical calculations of these quantities are not possible due to the lack of complete data we consider current status information which represents a more severe form of censoring than the commonly used right censoring asymptotic validity of our estimators can be justified using consistency results for nonparametric regression estimators finite sample behavior of our estimators is studied by simulation in which we show that our estimators based on these limited data compare well with those based on complete data we also apply our method to a real life data set arising from a cardiovascular diseases study in taiwan

fault tolerance in noise enhanced propagation as a mock up of synaptic transmission between neurons we revisit a problem that has recently risen the interest of several authors the propagation of a low frequency periodic signal through a chain of one way coupled bistable oscillators subject to uncorrelated additive noise on a numerical study performed in the optimal range of noise intensity for which essentially undamped transmission along the chain has been reported we focus on the outcome of feeding with noise all the nodes but the central one a moderate critical value of the coupling between oscillators is found such that whereas below it the propagation can be considered to be interrupted at the dead neuron it is reestablished above it thus one of the distinguishing features of synaptic transmission namely a fault tolerant behavior that enhances reliability at the expense of efficiency arises here as an emergent property of the system

monotone nonparametric regression in monotone regression procedures one utilizes only the monotonicity of the regression function in nonparametric regression one utilizes only the assumed smoothness the analytic and asymptotic properties of the estimator are superior in the latter case however monotonicity is not guaranteed we study a hybrid procedure that produces monotone estimators with properties similar to those of nonparametric regression estimators

uncertainty relationships in many body systems rigorous relationships among radial and logarithmic expectation values of an n body system are obtained by means of information theoretic methods especially interesting are the uncertainty expressions in terms of the product r n pb p which generalize the well known relation r2 p2 2 9n2 4 additionally an inequality involving the uncertainty in the logarithmic radius and the logarithmic momentum is shown for illustration the aocmcy of the inequalities found are numerically analysed in a hameefock framework for atomic systems finally it is pointed out how these inequalities are improved in atoms by raking into account other physical quantities or by considering some semiempirical results on atomic information entropies the one particle density p r is a fundamental quantity in the study of the physical and chemical properties of many fermion systems parr and yang 1989 since hohenberg and kohn 1964 formulated their theorem about the existence of a universal functional e pl for the energy of the many fermion system much effort has been paid in the study of the function p

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

dose response curve estimation a semiparametric mixture approach summary in the estimation of a dose response curve parametric models are straightforward and efficient but subject to model misspecifications nonparametric methods are robust but less efficient as a compromise we propose a semiparametric approach that combines the advantages of parametric and nonparametric curve estimates in a mixture form our estimator takes a weighted average of the parametric and nonparametric curve estimates in which a higher weight is assigned to the estimate with a better model fit when the parametric model assumption holds the semiparametric curve estimate converges to the parametric estimate and thus achieves high efficiency when the parametric model is misspecified the semiparametric estimate converges to the nonparametric estimate and remains consistent we also consider an adaptive weighting scheme to allow the weight to vary according to the local fit of the models we conduct extensive simulation studies to investigate the performance of the proposed methods and illustrate them with two real examples

likelihood based inference for monotone response models the behavior of maximum likelihood estimates mles and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied this class of problems differs radically from the usual parametric or semiparametric situations in that the mle of the monotone function at a point converges to the truth at rate n 1 3 slower than the usual n rate with a non gaussian limit distribution a framework for likelihood based estimation of monotone functions is developed and limit theorems describing the behavior of the mles and the likelihood ratio statistic are established in particular the likelihood ratio statistic is found to be asymptotically pivotal with a limit distribution that is no longer 2 but can be explicitly characterized in terms of a functional of brownian motion applications of the main results are presented and potential extensions discussed

monotone regression estimates for grouped observations the maximum likelihood estimator of a nondecreasing regression function with normally distributed errors has been considered in the literature its asymptotic distribution at a point is related to a solution of the heat equation and its rate of convergence to the underlying regression function is of order n 1 3 this estimator can be modified by grouping adjacent observations and then isotonizing the corresponding means it is shown that the resulting estimator has an asymptotic normal distribution for certain group sizes and its rate of convergence is of order n 215 the results of a simulation study for small sample sizes are presented and grouping procedures are discussed

bayesian monotone regression using gaussian process projection shape constrained regression analysis has applications in dose response modelling environmental risk assessment disease screening and many other areas incorporating the shape constraints can improve estimation efficiency and avoid implausible results we propose a novel method focusing on monotone curve and surface estimation which uses gaussian process projections our inference is based on projecting posterior samples from the gaussian process we develop theory on continuity of the projection and rates of contraction our approach leads to simple computation with good performance in finite samples the proposed projection method can also be applied to other constrained function estimation problems including those in multivariate settings

nonparametric bayesian bioassay including ordered polytomous response abstract previous attempts at implementing fully bayesian nonparametric bioassay have enjoyed limited success due to computational difficulties we show here how this problem may be generally handled using a sampling based approach to develop desired marginal posterior distributions and their features a useful extension is presented which treats the case of ordered polytomous response illustrative examples are provided

nonparametric analysis of dependently interval censored failure time data failure time studies based on observational cohorts often have to deal with irregular intermittent observation of individuals which produces interval censored failure times when the observation times depend on factors related to a person s failure time the failure times may be dependently interval censored inverse intensity of visit weighting methods have been developed for irregularly observed longitudinal or repeated measures data and recently extended to parametric failure time analysis this article develops nonparametric estimation of failure time distributions using weighted generalized estimating equations and monotone smoothing techniques simulations are conducted for examination of the finite sample performance of proposed estimators this research is motivated in part by the toronto psoriatic arthritis cohort study and the proposed methodology is applied to this study

the monotone smoothing of scatterplots we consider the problem of summarizing a scatterplot with a smooth monotone curve a solution that combines local averaging and isotonic regression is proposed and we demonstrate its use with two examples in the second example the procedure is applied in a regression setting to some data from box and cox 1964 and it is shown how this new procedure generalizes box and cox s well known family of transformations in the same example the bootstrap is applied to obtain a measure of the variability of the procedure

distribution free tests of stochastic monotonicity this article proposes an omnibus test for monotonicity of nonpara metric conditional distributions and its moments unlike previous proposals our method does not require smooth estimation of the derivatives of nonparametric curves and it can be implemented even when probability densities do not exist in fact we only require continuity of the marginal distributions under the null and xed alternatives distinguishing features of our approach are that critical values are pivotal under the null in nite samples and the test is invariant to any monotonic continuous transformation of the explanatory variable the test sta tistic is the sup norm of the dierence between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate the resulting test is able to detect local alternatives converging to the null at the parametric rate n 1 2 with n the sample size the article also discusses several applications and extensions of the proposal these include testing monotonicity of general conditional moments and the extension to multivariate explanatory variables the nite sample performance of the test is examined by means of a monte carlo experiment

tests for monotonicity of a regression mean with guaranteed level in this paper a nonparametric procedure for testing for monotonicity of a regression mean with guaranteed level is proposed the procedure is based on signs of differences of observations from the response variable the test is calibrated against the most difficult null hypothesis when the regression function is constant and produces an exact test in this context in general the test is conservative the power of the test is good and comparable with that of other nonparametric tests it is shown that the testing procedure has asymptotic power 1 against certain local alternatives the method is also robust against heavy tailed error distributions and even maintains good power when the errors are for example cauchy distributed a simulation study is provided to demonstrate finite sample behaviour of the testing procedure

a computational framework for multivariate convex regression and its variants we study the nonparametric least squares estimator lse of a multivariate convex regression function the lse given as the solution to a quadratic program with o n2 linear constraints n being t

classical laplace estimation for n 3 consistent estimators improved convergence rates and rate adaptive inference abstract we propose a classical laplace estimator alternative for a large class of n 3 consistent estimators including isotonic regression monotone hazard and maximum score estimators the proposed alternative provides a unified method of smoothing easier computation is a byproduct in the maximum score case depending on input parameter choice and smoothness the convergence rate of our estimator varies between n 3 and almost n and its limit distribution varies from chernoff to normal we provide a bias reduction method and an inference procedure which automatically adapts to the correct convergence rate and limit distribution