some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

an exact likelihood analysis of the multinomial probit model abstract we develop new methods for conducting a finite sample likelihood based analysis of the multinomial probit model using a variant of the gibbs sampler an algorithm is developed to draw from the exact posterior of the multinomial probit model with correlated errors this approach avoids direct evaluation of the likelihood and thus avoids the problems associated with calculating choice probabilities which affect both the standard likelihood and method of simulated moments approaches both simulated and actual consumer panel data are used to fit six dimensional choice models we also develop methods for analyzing random coefficient and multiperiod probit models

generalized linear array models with applications to multidimensional smoothing summary data with an array structure are common in statistics and the design or regression matrix for analysis of such data can often be written as a kronecker product factorial designs contingency tables and smoothing of data on multidimensional grids are three such general classes of data and models in such a setting we develop an arithmetic of arrays which allows us to define the expectation of the data array as a sequence of nested matrix operations on a coefficient array we show how this arithmetic leads to low storage high speed computation in the scoring algorithm of the generalized linear model we refer to a generalized linear array model and apply the methodology to the smoothing of multidimensional arrays we illustrate our procedure with the analysis of three data sets mortality data indexed by age at death and year of death spatially varying microarray background data and disease incidence data indexed by age at death year of death and month of death

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

smoothing spline gaussian regression more scalable computation via efficient approximation summary smoothing splines via the penalized least squares method provide versatile and effective nonparametric models for regression with gaussian responses the computation of smoothing splines is generally of the order o n3 n being the sample size which severely limits its practical applicability we study more scalable computation of smoothing spline regression via certain low dimensional approximations that are asymptotically as efficient a simple algorithm is presented and the bayes model that is associated with the approximations is derived with the latter guiding the porting of bayesian confidence intervals the practical choice of the dimension of the approximating space is determined through simulation studies and empirical comparisons of the approximations with the exact solution are presented also evaluated is a simple modification of the generalized cross validation method for smoothing parameter selection which to a large extent fixes the occasional undersmoothing problem that is suffered by generalized cross validation

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

the estimation of residual variance in nonparametric regression summary a wide class of estimators of the residual variance in nonparametric regression is considered namely those that are quadratic in the data unbiased for linear regression and always nonnegative the minimax mean squared error estimator over a natural class of regression functions is derived this optimal estimator has an interesting structure and is closely related to a minimax estimator of the regression curve itself

evaluation of the undetected error probability of single parity check product codes in this paper an expression for the undetected error probability p epsilon of single parity check product spcp codes used for error detection over a binary symmetric channel is derived it is shown that square spcp codes need not obey a certain commonly used bound approximate expressions for the maximum p epsilon and the corresponding maximizing e are given

poor error correction codes are poor error detection codes corresp we show that binary group codes that do not satisfy the asymptotic varshamov gilbert bound have an undesirable characteristic when used as error detection codes for transmission over the binary symmetric channel

nonparametric estimation of residual variance revisited summary several difference based estimators of residual variance are compared for finite sample size since the introduction of a rather simple estimator by gasser sroka jennensteinmetz 1986 other proposals have been made here the one given by hall kay titterington 1990 is of particular interest it minimizes the asymptotic variance unfortunately it has severe problems with finite sample bias and the estimator of gasser et al 1986 proves still to be a good choice a new estimator is introduced compromising between bias and variance

smoothing spline estimation in varying coefficient models summary smoothing spline estimators are considered for inference in varying coefficient models with one effect modifying covariate bayesian confidence intervals are developed for the coefficient curves and efficient computational methods are derived for computing the curve estimators fitted values posterior variances and data adaptive methods for selecting the levels of smoothing the efficacy and utility of the methodology proposed are demonstrated through a small simulation study and the analysis of a real data set

fitting multivariate regression functions by interaction spline models in nonparametric multivariate regression analysis we seek methods to reduce the dimensionality of the regression function to bypass the difficulty caused by the curse of dimensionality the original additive models approximate a regression function by additive univariate functions which overcome successfully the difficulty caused by the curse of dimensionality and have a clear interpretation but since they ignore any interactions between regressors they are not sufficiently flexible to cope with the case where interactions are of concern

on gibbs sampling for state space models summary we show how to use the gibbs sampler to carry out bayesian inference on a linear state space model with errors that are a mixture of normals and coefficients that can switch over time our approach simultaneously generates the whole of the state vector given the mixture and coefficient indicator variables and simultaneously generates all the indicator variables conditional on the state vectors the states are generated efficiently using the kalman filter we illustrate our approach by several examples and empirically compare its performance to another gibbs sampler where the states are generated one at a time the empirical results suggest that our approach is both practical to implement and dominates the gibbs sampler that generates the states one at a time

semiparametric bayesian inference in smooth coefficient models abstract we describe procedures for bayesian estimation and testing in cross sectional panel data and nonlinear smooth coefficient models the smooth coefficient model is a generalization of the partially linear or additive model wherein coefficients on linear explanatory variables are treated as unknown functions of an observable covariate in the approach we describe points on the regression lines are regarded as unknown parameters and priors are placed on differences between adjacent points to introduce the potential for smoothing the curves the algorithms we describe are quite simple to implement for example estimation testing and smoothing parameter selection can be carried out analytically in the cross sectional smooth coefficient model we apply our methods using data from the national longitudinal survey of youth nlsy using the nlsy data we first explore the relationship between ability and log wages and flexibly model how returns to schooling vary with measured cognitive ability we also examine a model of female labor supply and use this example to illustrate how the described techniques can been applied in nonlinear settings

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

functional mixed effects models in this article a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced the linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects which are modeled as realizations of zero mean stochastic processes the fixed functional effects and the random functional effects are modeled in the same functional space which guarantee the population average and subject specific curves have the same smoothness property these models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures can include continuous covariates as well as dummy factors in both the fixed or random design matrices and include the nested curves models as special cases two estimation procedures are proposed the first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software the second procedure is a sequential estimation procedure using kalman filtering this algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets a generalized maximum likelihood gml ratio test is proposed for inference and model selection an application to comparison of cortisol profiles is used as an illustration

automated smoothing of image and other regularly spaced data this paper is primarily motivated by the problem of automatically removing unwanted noise from high dimensional remote sensing imagery the initial step involves the transformation of the data to a space of intrinsically lower dimensionality and the smoothing of images in the new space different images require different amounts of smoothing the signal assumed to be mostly smooth with relatively few discontinuities is estimated from the data using the method of generalized cross validation it is shown how the generalized cross validated thin plate smoothing spline with observations on a regular grid in d dimensions is easily approximated and computed in the fourier domain space domain approximations are also investigated the technique is applied to some remote sensing data

on the probability of undetected error for linear block codes the problem of computing the probability of undetected error is considered for linear block codes used for error detection the recent literature is first reviewed and several results are extended it is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual using the dual code formulation the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes several bounds on the undetected error probability are then presented we conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained an efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results

when does more regularization imply fewer degrees of freedom sufficient conditions and counterexamples regularization aims to improve prediction performance by trading an increase in training error for better agreement between training and prediction errors which is often captured through decreased degrees of freedom in this paper we give examples which show that regularization can increase the degrees of freedom in common models including the lasso and ridge regression in such situations both training error and degrees of freedom increase making the regularization inherently without merit two important scenarios are described where the expected reduction in degrees of freedom is guaranteed all symmetric linear smoothers and convex constrained linear regression models like ridge regression and the lasso when compared to unconstrained linear regression

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

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series

generalized coorbit theory banach frames and the relation to α modulation spaces this paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients that has been developed quite recently we show that the general theory applied to the affine weyl heisenberg group gives rise to families of smoothness spaces that can be identified with modulation spaces

optimal spline smoothing of fmri time series by generalized cross validation linear parametric regression models of fmri time series have correlated residuals one approach to address this problem is to condition the autocorrelation structure by temporal smoothing smoothing splines with the degree of smoothing selected by generalized cross validation gcv spline provide a method to find an optimal smoother for an fmri time series the purpose of this study was to determine if gcv spline of fmri time series yields unbiased variance estimates of linear regression model parameters gcv spline was evaluated with a real fmri data set and bias of the variance estimator was computed for simulated time series with autocorrelation structures derived from fmri data this study only considered fmri experimental designs of boxcar type the results from the real data suggest that gcv spline determines appropriate amounts of smoothing the simulations show that the variance estimates are on average unbiased the unbiased variance estimates come at some cost to signal detection efficiency this study demonstrates that gcv spline is an appropriate method for smoothing fmri time series

confidence intervals for nonparametric curve estimates numerous nonparametric regression methods exist that yield consistent estimators of function curves often one is also interested in constructing confidence intervals for the unknown function when a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 on average but because bias in nonparametric estimation is not uniform they do not hold the desired level uniformly at all design points most research in this area has focused on mean squared error properties of the estimator for example mise itself a global measure in addition measures like mise are one step removed from the practical issue of coverage probability recent work that focuses on coverage probability has considered only coverage in an average sense ignoring the important issue of uniformity of coverage across the design space to deal with this problem a new estimator is developed which uses a local cross validation criterion lcv to determi

penalized clustering of large scale functional data with multiple covariates in this article we propose a penalized clustering method for large scale data with multiple covariates through a functional data approach in our proposed method responses and covariates are linked together through nonparametric multivariate functions fixed effects which have great flexibility in modeling various function features such as jump points branching and periodicity functional anova is used to further decompose multivariate functions in a reproducing kernel hilbert space and provide associated notions of main effect and interaction parsimonious random effects are used to capture various correlation structures the mixed effects models are nested under a general mixture model in which the heterogeneity of functional data is characterized we propose a penalized henderson s likelihood approach for model fitting and design a rejection controlled em algorithm for the estimation our method selects smoothing parameters through generalized cross validation furthermore bayesian confidence in

the affine uncertainty principle in one and two dimensions abstract in this paper we construct families of wavelets that minimize an uncertainty relation associated with square integrable representations of some canonical groups especially we obtain a new interpretation of the mexican hat function

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

an iterative approach to two dimensional laplacian smoothing with application to image restoration abstract smoothing methods are increasingly important in modern graphically oriented data analysis there are several fast and reliable univariate cross validated smoothing algorithms but comparable methods are not available in higher dimensions this article develops an iterative approach to two dimensional laplacian spline smoothing based on tensor product cubic b splines estimating equations are developed and solved iteratively using two line symmetric successive over relaxation with conjugate gradient acceleration an asymptotic approximation for the generalized cross validation score is described the resulting algorithm is naturally suited to two dimensional scatterplot smoothing but has potential value for image restoration as well this is illustrated on a multicolor image example along with an image restoration an uncertainty assessment based on the entropy of the marginal posterior pixel value distribution is produced this analysis particularly highlights the statistical uncertainty in resol

nonparametric input estimation in physiological systems problems methods and case studies input estimation from output data is an important problem in the analysis of physiological systems because many signals of interest are not directly accessible to measurement when the system is time invariant this problem is often referred to as deconvolution three representative physiological problems regarding hormone secretion insulin dynamics and hepatic glucose production are used to illustrate the major challenges ill conditioning confidence intervals assessment infrequent and nonuniform sampling nonnegativity constraints and computational efficiency the paper provides a critical overview of the existing techniques focusing on regularization theory and bayesian estimation in order to overcome some inadequacies of the existing methods some new results are derived in particular the connection between the maximum likelihood estimate of the regularization parameter and the notion of equivalent degree of freedom is studied moreover a fast svd based numerical algorithm is developed that includes the optimization of the regularization parameter and the computation of confidence intervals the proposed techniques are validated on a benchmark problem and are shown to provide effective solutions to the three physiological case studies

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

efficient computation of smoothing splines via adaptive basis sampling summary smoothing splines provide flexible nonparametric regression estimators however the high computational cost of smoothing splines for large datasets has hindered their wide application in this article we develop a new method named adaptive basis sampling for efficient computation of smoothing splines in super large samples except for the univariate case where the reinsch algorithm is applicable a smoothing spline for a regression problem with sample size n can be expressed as a linear combination of n basis functions and its computational complexity is generally o n 3 we achieve a more scalable computation in the multivariate case by evaluating the smoothing spline using a smaller set of basis functions obtained by an adaptive sampling scheme that uses values of the response variable our asymptotic analysis shows that smoothing splines computed via adaptive basis sampling converge to the true function at the same rate as full basis smoothing splines using simulation studies and a large scale deep earth core mantle boundary imaging study we show that the proposed method outperforms a sampling method that does not use the values of response variables

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

a comparison of the reinsch and speckman splines in the context of nonparametric regression spline smoothers can be used both for the estimation of the curve itself and for the estimation of the error variance we compare two approaches that have been suggested the reinsch spline i e roughness penalty and the speckman spline minimax the comparison is made on the basis of the summed mean squared error for curve estimation and on the basis of a closely related criterion for variance estimation we show in both asymptotic and small sample studies that the reinsch smoothing spline is close to optimal both as a curve and as a variance estimator

a bayesian committee machine the bayesian committee machine bcm is a novel approach to combining estimators that were trained on different data sets although the bcm can be applied to the combination of any kind of estimators the main foci are gaussian process regression and related systems such as regularization networks and smoothing splines for which the degrees of freedom increase with the number of training data somewhat surprisingly we find that the performance of the bcm improves if several test points are queried at the same time and is optimal if the number of test points is at least as large as the degrees of freedom of the estimator the bcm also provides a new solution for on line learning with potential applications to data mining we apply the bcm to systems with fixed basis functions and discuss its relationship to gaussian process regression finally we show how the ideas behind the bcm can be applied in a non bayesian setting to extend the input dependent combination of estimators

smoothing spline analysis of variance models a new tool for the analysis of cyclic biomechanical data cyclic biomechanical data are commonplace in orthopedic rehabilitation and sports research where the goal is to understand and compare biomechanical differences between experimental conditions and or subject populations a common approach to analyzing cyclic biomechanical data involves averaging the biomechanical signals across cycle replications and then comparing mean differences at specific points of the cycle this pointwise analysis approach ignores the functional nature of the data which can hinder one s ability to find subtle differences between experimental conditions and or subject populations to overcome this limitation we propose using mixed effects smoothing spline analysis of variance ssanova to analyze differences in cyclic biomechanical data the ssanova framework makes it possible to decompose the estimated function into the portion that is common across groups i e the average cycle ac and the portion that differs across groups i e the contrast cycle cc by partitioning the signal in such a manner we can obtain estimates of the cc differences ccds which are the functions directly describing group differences in the cyclic biomechanical data using both simulated and experimental data we illustrate the benefits of using ssanova models to analyze differences in noisy biomechanical gait signals collected from multiple locations joints of subjects participating in different experimental conditions using bayesian confidence intervals the ssanova results can be used in clinical and research settings to reliably quantify biomechanical differences between experimental conditions and or subject populations

a statistical model for signature verification a bayesian model for off line signature verification involving the representation of a signature through its curvature is developed the prior model makes use of a spatial point process for specifying the knots in an approximation restricted to a buffer region close to a template curvature along with an independent time warping mechanism in this way prior shape information about the signature can be built into the analysis the observation model is based on additive white noise superimposed on the underlying curvature the approach is implemented using markov chain monte carlo and applied to a collection of documented instances of william shakespeare s signature

bayesian model discrimination and bayes factors for linear gaussian state space models summary it is shown how to discriminate between different linear gaussian state space models for a given time series by means of a bayesian approach which chooses the model that minimizes the expected loss a practical implementation of this procedure requires a fully bayesian analysis for both the state vector and the unknown hyperparameters and is carried out by markov chain monte carlo methods an application to some non standard situations such as testing hypotheses on the boundary of the parameter space discriminating non nested models and discrimination of more than two models is discussed in detail

adaptive confidence bands for nonparametric regression functions this article proposes a new formulation for the construction of adaptive confidence bands cbs in nonparametric function estimation problems cbs which have size that adapts to the smoothness of the function while guaranteeing that both the relative excess mass of the function lying outside the band and the measure of the set of points where the function lies outside the band are small it is shown that the bands adapt over a maximum range of lipschitz classes the adaptive cb can be easily implemented in standard statistical software with wavelet support we investigate the numerical performance of the procedure using both simulated and real datasets the numerical results agree well with the theoretical analysis the procedure can be easily modified and used for other nonparametric function estimation models supplementary materials for this article are available online

smoothing spline anova for time dependent spectral analysis in this article we propose a smoothing spline anova model ss anova to estimate and to make inference on the time varying log spectrum of a locally stationary process the time varying spectrum is assumed to be smooth in both time and frequency this assumption essentially turns a time frequency spectral estimation problem into a 2 dimensional surface estimation problem a smooth localized complex exponential slex basis is used to calculate the initial periodograms and a ss anova is fitted to the log periodograms this approach allows the time and frequency domains to be modeled in a unified approach and jointly estimated inference procedures such as confidence intervals and hypothesis tests proposed for the ss anova can be adopted for the time varying spectrum because of the smoothness assumption of the underlying spectrum once we have the estimates on a time frequency grid we can calculate the estimate at any given time and frequency this leads to a high computational efficiency because for

estimation filtering and smoothing in state space models with incompletely specified initial conditions on definit la vraisemblance pour un modele d espace d etats avec des conditions initiales incompletement specifiees par transformation des donnees pour eliminer la dependance sur des conditions non specifiees on etend cette approche pour obtenir des estimations sur les vecteurs d etat les predicteurs et les interpolateurs pour des observations manquantes

a probabilistic approach to pattern matching in the continuous domain the goal of this paper is to solve the following basic problem given discrete noisy samples from a continuous signal compute the probability distribution of its distance from a fixed template as opposed to the typical restoration problem which considers a single optimal signal the computation of the entire probability distribution necessitates integrating over the entire signal space to achieve this we apply path integration techniques the problem is studied in one and two dimensions and an accurate solution as well as an efficient approximation scheme are provided

a note on smoothness priors and nonlinear regression abstract a generalization of a nonparametric estimator proposed by shiller 1984 is considered the estimator is shown to be directly related to penalized least squares estimation and spline smoothing some simplifications concerning the computation of shiller s estimator and corresponding posterior covariances are indicated

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

testing for integration using evolving trend and seasonals models a bayesian approach in this paper we make use of state space models to investigate the presence of stochastic trends in economic time series a model is specified where such a trend can enter either in the autoregressive representation or in a separate state equation tests based on the former are analogous to dickey fuller tests of unit roots while the latter are analogous to kpss tests of trend stationarity we use bayesian methods to survey the properties of the likelihood function in such models and to calculate posterior odds ratios comparing models with and without stochastic trends we extend these ideas to the problem of testing for integration at seasonal frequencies and show how our techniques can be used to carry out bayesian variants of either the hegy or canova hansen test stochastic integration rules based on markov chain monte carlo as well as deterministic integration rules are used strengths and weaknesses of each approach are indicated

bayesian confidence intervals for smoothing splines abstract the frequency properties of wahba s bayesian confidence intervals for smoothing splines are investigated by a large sample approximation and by a simulation study when the coverage probabilities for these pointwise confidence intervals are averaged across the observation points the average coverage probability acp should be close to the nominal level from a frequency point of view this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error these properties are independent of the bayesian assumptions used to derive this confidence procedure and they explain why the acp is accurate for functions that are much smoother than the sample paths prescribed by the prior this analysis accounts for the choice of the smoothing parameter bandwidth using cross validation in the case of natural splines an adaptive method for avo

existence and computation of optimally localized coherent states this paper is concerned with localization properties of coherent states instead of classical uncertainty relations we consider generalized localization quantities this is done by introducing measures on the reproducing kernel in this context we may prove the existence of optimally localized states moreover we provide a numerical scheme for deriving them

on p values for smooth components of an extended generalized additive model the problem of testing smooth components of an extended generalized additive model for equality to zero is considered confidence intervals for such components exhibit good across the function coverage probabilities if based on the approximate result hat f i n f i v f i i where f is the vector of evaluated values for the smooth component of interest and v f is the covariance matrix for f according to the bayesian view of the smoothing process based on this result a wald type test of f 0 is proposed it is shown that care must be taken in selecting the rank used in the test statistic the method complements previous work by extending applicability beyond the gaussian case while considering tests of zero effect rather than testing the parametric hypothesis given by the null space of the component s smoothing penalty the proposed p values are routine and efficient to compute from a fitted model without requiring extra model fits or null distribution simulation

semiparametric mixed models for increment averaged data with application to carbon sequestration in agricultural soils adoption of conservation tillage practice in agriculture offers the potential to mitigate greenhouse gas emissions studies comparing conservation tillage methods to traditional tillage pair fields under the two management systems and obtain soil core samples from each treatment cores are divided into multiple increments and matching increments from one or more cores are aggregated and analyzed for carbon stock these data represent not the actual value at a specific depth but rather the total or average over a depth increment a semiparametric mixed model is developed for such increment averaged data the model uses parametric fixed effects to represent covariate effects random effects to capture correlation within studies and an integrated smooth function to describe effects of depth the depth function is specified as an additive model estimated with penalized splines using standard mixed model software smoothing parameters are automatically selected using restricted maximum likelihood the meth

adaptive confidence bands we show that there do not exist adaptive confidence bands for curve estimation except under very restrictive assumptions we propose instead to construct adaptive bands that cover a surrogate function f which is close to but simpler than f the surrogate captures the significant features in f we establish lower bounds on the width for any confidence band for f and construct a procedure that comes within a small constant factor of attaining the lower bound for finite samples

statistical inference in the multinomial multiperiod probit model abstract statistical inference in multinomial multiperiod probit models has been hindered in the past by the high dimensional numerical integrations necessary to form the likelihood functions posterior distributions or moment conditions in these models we describe three alternative estimators implemented using simulation based approaches to inference that circumvent the integration problem posterior means computed using gibbs sampling and data augmentation gibbs simulated maximum likelihood sml estimation using the ghk probability simulator and method of simulated moment msm estimation using ghk we perform a set of monte carlo experiments to compare the sampling distributions of these estimators although all three estimators perform reasonably well some important differences emerge our most important finding is that holding simulation size fixed the relative and absolute performance of the classical methods especially sml gets worse when serial correlation in disturbances is strong in data sets with an ar 1 parameter of 0 50 the rmses for sml and msm based on ghk with 20 draws exceed those of gibbs by 9 and 0 respectively but when the ar 1 parameter is 0 80 the rmses for sml and msm based on 20 draws exceed those of gibbs by 79 and 37 respectively and the number of draws needed to reduce the rmses to within 10 of gibbs are 160 and 80 respectively also the sml estimates of serial correlation parameters exhibit significant downward bias thus while conventional wisdom suggests that 20 draws of ghk is enough to render the bias and noise induced by simulation negligible our results suggest that much larger simulation sizes are needed when serial correlation in disturbances is strong

smoothing noisy data with spline functions a procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m 1 fitted ton distinct not necessarily equally spaced or uniformly weighted data points is presented the procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline including the generalized cross validation the method is a significant improvement over an existing method which requires ordern 3 operations

more results on the weight enumerator of product codes we consider the product code c sub p of q ary linear codes with minimum distances d sub c and d sub r the words in c sub p of weight less than d sub r d sub c max d sub r spl lceil d sub c g spl rceil d sub c spl lceil d sub r q spl rceil are characterized and their number is expressed in the number of low weight words of the constituent codes for binary product codes we give an upper bound on the number of words in c sub p of weightless than min d sub r d sub c spl lceil d sub c 2 spl rceil 1 d sub c d sub r spl lceil d sub r 2 spl rceil 1 that is met with equality if c sub c and c sub r are extended perfect codes

bayesian variants of some classical semiparametric regression techniques abstract this paper develops new bayesian methods for semiparametric inference in the partial linear normal regression model y z f x e where f is an unknown function these methods draw solely on the normal linear regression model with natural conjugate prior hence posterior results are available which do not suffer from some problems which plague the existing literature such as computational complexity methods for testing parametric regression models against semiparametric alternatives are developed we discuss how these methods can at some cost in terms of computational complexity be extended to other models e g qualitative choice models or those involving censoring or truncation and provide precise details for a semiparametric probit model we show how the assumption of normal errors can easily be relaxed

hierarchical analysis of sur models with extensions to correlated serial errors and time varying parameter models we consider the use of markov chain monte carlo methods to analyze hierarchical versions of zellner s sur model in this context the questions of bayes estimation and model adequacy checking are considered the approach is extended to sur model with vector autoregressive and vector moving average errors of the first order finally an efficient algorithm is developed to estimate a markov time varying parameter sur model the ideas are applied to both simulated and real data

on testing for improper error detection codes a family of tests for improper codes is given these tests can be used in cases where the complete weight distribution of the code is unknown it was found that knowledge of the number of minimum weight codewords can be used to greatly increase the effectiveness of the asymptotic varshamov gilbert test further improvement is possible as more is known about the number of other weight codewords

a bayesian approach to model selection in stochastic coefficient regression models and structural time series models abstract a bayesian model selection procedure is proposed for a stochastic coefficient regression model to determine which coefficients are fixed and which are time varying the posterior probabilities are computed by gaussian quadrature using the kalman filter it is shown empirically that the model selection approach works well on both simulated and real data a similar approach can be used to select a model from a class of state space models in particular for a trend plus seasonal structural time series model we show how to determine if the trend and or seasonal component is deterministic or stochastic

an efficient method for calculating smoothing splines using orthogonal transformations a procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m 1 using an orthogonal factorization is presented the procedure substantially overcomes the problem of ill conditioning encountered by a recently developed method which employs a cholesky factorization but still requires only orderm 2 n operations and ordermn storage

a signal extraction approach to modeling hormone time series with pulses and a changing baseline abstract hormones serve as regulating signals for many biological processes in recent years it was determined that many hormones are secreted in a pulsatile manner and that the pulsatile secretion pattern in addition to the absolute concentration level is important in regulating biological processes consequently it is necessary to characterize the latent secretion patterns from measurements of concentration levels the characterization is complicated by the presence of a biological circadian rhythm when hormone concentrations are plotted over time the resultant time series usually exhibits occasional short rises superimposed on a slowly changing baseline this is a result of a mixture of pulsatile secretions and a circadian rhythm in this article we present a signal extraction approach to model simultaneously a slowly changing component and a pulsatile component of a time series a smoothing spline is used to model the baseline and a multiprocess dynamic linear model is used to model the pulsati

a note upon another error detecting code that is not proper in this correspondence the performance of the nonlinear error detecting code described in ccir recommendation 476 2 when used over a memoryless binary symmetric channel bsc is derived from this it is apparent that the probability of accepting an erroneous block is not a monotonic function of the bit error probability but rather reaches a peak of 5 653 percent when the bit error probability is 11 66 percent consequently this code is not proper 1 6

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea