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

on bayesian consistency we consider a sequence of posterior distributions based on a data dependent prior which we shall refer to as a pseudoposterior distribution and establish simple conditions under which the sequence is hellinger consistent it is shown how investigations into these pseudo posteriors assist with the understanding of some true posterior distributions including polya trees the infinite dimensional exponential family and mixture models

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

results on the associated jacobi and gegenbauer polynomials abstract recurrence relations for the coefficients of the jacobi polynomial form of the associated jacobi polynomials p n x c are given for certain values of explicit formulae for these coefficients are obtained in particular the classical watson formula for the first associated gegenbauer polynomials is generalized

nonparametric function estimation subject to monotonicity convexity and other shape constraints this paper uses free knot and fixed knot regression splines in a bayesian context to develop methods for the nonparametric estimation of functions subject to shape constraints in models with log concave likelihood functions the shape constraints we consider include monotonicity convexity and functions with a single minimum a computationally efficient mcmc sampling algorithm is developed that converges faster than previous methods for non gaussian models simulation results indicate the monotonically constrained function estimates have good small sample properties relative to i unconstrained function estimates and ii function estimates obtained from other constrained estimation methods when such methods exist also asymptotic results show the methodology provides consistent estimates for a large class of smooth functions two detailed illustrations exemplify the ideas

multivariate convex regression with adaptive partitioning we propose a new nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function convexity constraints are common in economics statistics operations research financial engineering and optimization but there is currently no multivariate method that is stable and computationally feasible for more than a few thousand observations we introduce convex adaptive partitioning cap which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions cap is a computationally efficient consistent method for convex regression we demonstrate empirical performance by comparing the performance of cap to other shape constrained and unconstrained regression methods for predicting weekly wages and value function approximation for pricing american basket options

results on the associated classical orthogonal polynomials abstract let p k x be any system of the classical orthogonal polynomials and let p k x c be the corresponding associated polynomials of order c c n second order recurrence relation in k is given for the connection coefficient a n 1 k c in p n 1 x c k 0 n 1 a n 1 k c p k x this result is obtained thanks to a new explicit form of the fourth order differential equation satisfied by p n 1 c

a bayesian approach to non parametric monotone function estimation the paper proposes two bayesian approaches to non parametric monotone function estimation the first approach uses a hierarchical bayes framework and a characterization of smooth monotone functions given by ramsay that allows unconstrained estimation the second approach uses a bayesian regression spline model of smith and kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate the small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods asymptotic results are also given that show that bayesian methods provide consistent function estimators for a large class of smooth functions an example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple regression model where two functions are constrained to be monotone

nonparametric regression using bayesian variable selection abstract this paper estimates an additive model semiparametrically while automatically selecting the significant independent variables and the appropriate power transformation of the dependent variable the nonlinear variables are modeled as regression splines with significant knots selected from a large number of candidate knots the estimation is made robust by modeling the errors as a mixture of normals a bayesian approach is used to select the significant knots the power transformation and to identify outliers using the gibbs sampler to carry out the computation empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares favorably with a kernel weighted local linear smoother the variable selection algorithm in the paper is substantially faster than previous bayesian variable selection algorithms

fourth order differential equation satisfied by the associated of any order of all classical orthogonal polynomials a study of their distribution of zeros abstract the first associated numerator polynomials of all classical orthogonal polynomials satisfy one fourth order differential equation valid for the four classical families but for the associated of arbitrary order the differential equations are only known separately in this work we introduce a program built in mathematica symbolic language which is able to construct the unique differential equation satisfied by the associated of any order of the classical class then we use this differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of the kth power of zeros which are closely related with the moments of such a distribution

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

bayesian curve fitting with free knot splines we describe a bayesian method for fitting curves to data drawn from an exponential family that uses splines for which the number and locations of knots are free parameters the method uses reversible jump markov chain monte carlo to change the knot configurations and a locality heuristic to speed up mixing for nonnormal models we approximate the integrated likelihood ratios needed to compute acceptance probabilities by using the bayesian information criterion bic under priors that make this approximation accurate our technique is based on a marginalised chain on the knot number and locations but we provide methods for inference about the regression coefficients and functions of them in both normal and nonnormal models simulation results suggest that the method performs well and we illustrate the method in two neuroscience applications

generalized monotonic regression using random change points we introduce a procedure for generalized monotonic curve fitting that is based on a bayesian analysis of the isotonic regression model conventional isotonic regression fits monotonically increasing step functions to data in our approach we treat the number and location of the steps as random for each step level we adopt the conjugate prior to the sampling distribution of the data as if the curve was unconstrained we then propose to use markov chain monte carlo simulation to draw samples from the unconstrained model space and retain only those samples for which the monotonic constraint holds the proportion of the samples collected for which the constraint holds can be used to provide a value for the weight of evidence in terms of bayes factors for monotonicity given the data using the samples probability statements can be made about other quantities of interest such as the number of change points in the data and posterior distributions on the location of the change points can be provided the method is illustrated throughout by a reanalysis of the leukaemia data studied by schell and singh

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

calibration and empirical bayes variable selection for the problem of variable selection for the normal linear model selection criteria such as aic c p bic and ric have fixed dimensionality penalties such criteria are shown to correspond to selection of maximum posterior models under implicit hyperparameter choices for a particular hierarchical bayes formulation based on this calibration we propose empirical bayes selection criteria that use hyperparameter estimates instead of fixed choices for obtaining these estimates both marginal and conditional maximum likelihood methods are considered as opposed to traditional fixed penalty criteria these empirical bayes criteria have dimensionality penalties that depend on the data their performance is seen to approximate adaptively the performance of the best fixed penalty criterion across a variety of orthogonal and nonorthogonal set ups including wavelet regression empirical bayes shrinkage estimators of the selected coefficients are also proposed

fourth order differential equations for numerator polynomials the author gives explicitly the fourth order differential equation satisfied by the numerator polynomials associated polynomials of the classical orthogonal polynomials the coefficients of the differential equation are at most a quadratic combination of the polynomials sigma and tau and their derivatives defined via the relation sigma rho te this differential equation is therefore valid for the jacobi laguerre hermite and bessel associated polynomials

on the use of non local prior densities in bayesian hypothesis tests we examine philosophical problems and sampling deficiencies that are associated with current bayesian hypothesis testing methodology paying particular attention to objective bayes methodology because the prior densities that are used to define alternative hypotheses in many bayesian tests assign non negligible probability to regions of the parameter space that are consistent with null hypotheses resulting tests provide exponential accumulation of evidence in favour of true alternative hypotheses but only sublinear accumulation of evidence in favour of true null hypotheses thus it is often impossible for such tests to provide strong evidence in favour of a true null hypothesis even when moderately large sample sizes have been obtained we review asymptotic convergence rates of bayes factors in testing precise null hypotheses and propose two new classes of prior densities that ameliorate the imbalance in convergence rates that is inherited by most bayesian tests using members of these classes we obtain analytic expressions for bayes factors in linear models and derive approximations to bayes factors in large sample settings

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

properties of the polynomials associated with the jacobi polynomials abstract power forms and jacobi polynomial forms are found for the polynomials w associated with jacobi polynomials also some differential difference equations and evalua tions of certain integrals involving wii 1 are given 1 introduction let w be a nonnegative function defined on an interval a b forwhich all momentsm j x w x dx n 0 1 exist and are finite m0 0 let be the monic polynomials orthogonal on a b with respect to w the polynomials 1 1 q x n pax rztt wu at pn t jt n 0 1 j a x are called the polynomials associated with the pn as is well known pn x a x are linearly independent solutions of the recurrence formula i 2 yn i x a y bny x q 0 1 where l3 an xpn p p pn n 0 o o b pn pn pn x pn x n l and f g f f x g x w x dx ja the initial conditions arep i x 0 pa x 1

testing convex hypotheses on the mean of a gaussian vector application to testing qualitative hypotheses on a regression function in this paper we propose a general methodology based on multiple testing for testing that the mean of a gaussian vector in r n belongs to a convex set we show that the test achieves its nominal level and characterize a class of vectors over which the tests achieve a prescribed power in the functional regression model this general methodology is applied to test some qualitative hypotheses on the regression function for example we test that the regression function is positive increasing convex or more generally satisfies a differential inequality uniform separation rates over classes of smooth functions are established and a comparison with other results in the literature is provided a simulation study evaluates some of the procedures for testing monotonicity

a consistent test of functional form via nonparametric estimation techniques abstract this paper presents a consistent test of functional form of nonlinear regression models the test combines the methodology of the conditional moment test and nonparametric estimation techniques using degenerate and nondegenerate u statistic theories the test statistic is shown to be asymptotically distributed standard normal under the null hypothesis that the parametric model is correct while diverging to infinity at a rate arbitrarily close to n the sample size if the parametric model is misspecified therefore the test is consistent against all deviations from the parametric model the test is robust to heteroskedasticity a version of the test can be constructed which will have asymptotic power equal to 1 against any local alternatives approaching the null at rates slower than the parametric rate 1 n a simulation study reveals that the test has good finite sample properties

the three term recurrence relation and the differentiation formulas for hypergeometric type functions abstract the functions of hypergeometric type are the solutions y y x of the differential equation z y z y y 0 where are polynomials of degrees not higher than 2 and 1 respectively and is a constant here we consider a class of functions of hypergeometric type with the additional condition that 1 2 1 0 being a complex number in general moreover we assume that the coefficients of the polynomials and have no dependence on to this class of functions belong gauss kummer and hermite functions the classical orthogonal polynomials and many other functions encountered in linear and non linear physics we obtain two important structural properties of these functions i the so called three term recurrence relation which correlates three functions of successive orders and ii the differentiation formulas also called ladder or structure relations or even differential recurrence relations which relate the first derivative y z with the functions y z and y 1 z or y 1 z finally these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y where is a positive integer number and the associated contour is closed for completeness the explicit expressions corresponding to all classical orthogonal polynomials jacobi laguerre hermite and bessel are tabulated

co recursive orthogonal polynomials and fourth order differential equations abstract we give explicitly the 4th order differential equation satisfied by the co recursive polynomials of the classical orthogonal polynomials we emphasize the legendre and bessel cases for which the co recursive differential equation coincide with the equation satisfied by the numerator polynomial differential equation and we propose new conjectures

rehabilitation of the gauss jordan algorithm in this paper a gauss jordan algorithm with column interchanges is presented and analysed we show that in contrast with gaussian elimination the gauss jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability for solutions obtained by gauss jordan with column interchanges a more satisfactory bound for the residual norm can be given the analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by gaussian elimination and that in most practical situations the residuals are equally small this is confirmed by numerical experiments moreover timing experiments on a cyber 205 vector computer show that the algorithm presented has good vectorisation properties

bayesian semiparametric isotonic regression for count data this article proposes a semiparametric bayesian approach for inference on an unknown isotonic regression function f x characterizing the relationship between a continuous predictor x and a count response variable y adjusting for covariates z a dirichlet process mixture of poisson distributions is used to avoid parametric assumptions on the conditional distribution of y given x and z then to also avoid parametric assumptions on f x a novel prior formulation is proposed that enforces the nondecreasing constraint and assigns positive prior probability to the null hypothesis of no association through the use of carefully tailored hyperprior distributions we allow for borrowing of information across different regions of x in estimating f x and in assessing hypotheses about local increases in the function due to conjugacy properties posterior computation is straightforward using a markov chain monte carlo algorithm the methods are illustrated using data from an epidemiologic study of sleep pr

nonparametric maximum likelihood estimation by the method of sieves maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space for example the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an operatorname iid sample the maximum of the likelihood is not attained by any density in this example as in many other examples the parameter space positive functions with area one is too big but the likelihood method can often be salvaged if we first maximize over a constrained subspace of the parameter space and then relax the constraint as the sample size grows this is grenander s method of sieves application of the method sometimes leads to new estimators for familiar problems or to a new motivation for an already well studied technique we will establish some general consistency results for the method and then we will focus on three applications

nonparametric bayesian testing for monotonicity this paper adopts a nonparametric bayesian approach to testing whether a function is monotone two new families of tests are constructed the first uses constrained smoothing splines with a hierarchical stochastic process prior that explicitly controls the prior probability of monotonicity the second uses regression splines together with two proposals for the prior over the regression coefficients via simulation the finite sample performance of the tests is shown to improve upon existing frequentist and bayesian methods the asymptotic properties of the bayes factor for comparing monotone versus nonmonotone regression functions in a gaussian model are also studied our results significantly extend those currently available which chiefly focus on determining the dimension of a parametric linear model

bayesian multivariate isotonic regression splines in many applications interest focuses on assessing the relationship between a predictor and a multivariate outcome variable and there may be prior knowledge about the shape of the regression curves for example regression functions that relate dose of a possible risk factor to different adverse outcomes can often be assumed to be nondecreasing in such cases interest focuses on 1 assessing evidence of an overall adverse effect 2 determining which outcomes are most affected and 3 estimating outcome specific regression curves this article proposes a bayesian approach for addressing this problem motivated by multisite tumor data from carcinogenicity experiments a multivariate smoothing spline model is specified that accommodates dependency in the multiple curves through a hierarchical markov random field prior for the basis coefficients while also allowing for residual correlation a gibbs sampler is proposed for posterior computation and the approach is applied to data on body weight and t