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 extended hazard model with longitudinal covariates summary in clinical trials and other medical studies it has become increasingly common to observe an event time of interest and longitudinal covariates simultaneously in the literature joint modelling approaches have been employed to analyze both survival and longitudinal processes and to investigate their association early attention has mostly been placed on developing adaptive and flexible longitudinal processes based on a prespecified survival model most commonly the cox proportional model in this paper we propose a general class of semi parametric hazard regres

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

toda type differential equations for the recurrence coefficients of orthogonal polynomials and freud transformation transformations of the measure of orthogonality for orthogonal polynomials namely freud transformations are considered jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations dynamics of the coefficients are described by generalized toda equations the classical toda lattice equations are the simplest special case of dynamics of the coefficients under the freud transformation of the measure of orthogonality also dynamics of hankel determinants its minors and other notions corresponding to the orthogonal polynomials are studied

interlacing of zeros of gegenbauer polynomials of non consecutive degree from different sequences a theorem due to stieltjes states that if p n n 0 infty is any orthogonal sequence then between any two consecutive zeros of p k there is at least one zero of p n whenever k frac 12 if 0 t k 1 and also to the zeros of c n 1 lambda and c n 2 lambda k if k in 1 2 3 more generally we prove that stieltjes interlacing holds between the zeros of the kth derivative of c n lambda and the zeros of c n 1 lambda k in 1 2 dots n 1 and we derive associated polynomials that play an analogous role to the de boor saff polynomials in completing the interlacing process of the zeros

an overall strategy based on regression models to estimate relative survival and model the effects of prognostic factors in cancer survival studies relative survival provides a measure of the proportion of patients dying from the disease under study without requiring the knowledge of the cause of death we propose an overall strategy based on regression models to estimate the relative survival and model the effects of potential prognostic factors the baseline hazard was modelled until 10 years follow up using parametric continuous functions six models including cubic regression splines were considered and the akaike information criterion was used to select the final model this approach yielded smooth and reliable estimates of mortality hazard and allowed us to deal with sparse data taking into account all the available information splines were also used to model simultaneously non linear effects of continuous covariates and time dependent hazard ratios this led to a graphical representation of the hazard ratio that can be useful for clinical interpretation estimates of these models were obtained by likelihood maximization we showed that these estimates could be also obtained using standard algorithms for poisson regression

smoothing splines regression derivatives and deconvolution abstract the statistical properties of a cubic smoothing spline and its derivative are analyzed it is shown that unless unnatural boundary conditions hold the integrated squared bias is dominated by local effects near the boundary similar effects are shown to occur in the regularized solution of a translation kernel integral equation these results are derived by developing a fourier representation for a smoothing spline author

hazard function estimation using b splines summary a flexible parametric procedure is given to model the hazard function as a linear combination of cubic b splines and to obtain maximum likelihood estimates from censored survival data the approach yields smooth estimates of the hazard and suivivorship functions that are intermediate in structure between strongly parametric and non parametric models a simple method is described for selecting the number and location of knots simulation results show favorable root mean square error compared to non parametric estimates for both the hazard and survivorship functions three methods are given to calculate confidence intervals based on the delta method profile likelihood and bootstrap respectively the procedure is applied to estimate hazard rates for acquired immunodeficiency syndrome aids following infection with human immunodeficiency virus hiv spline methods can accommodate complex censoring mechanisms such as those that arise in the aids setting to illustrate hiv infection incidence is estimated for a cohort of hemophiliacs in which the dates of hiv infection are interval censored and some subjects were born after the onset of the hiv epidemic this report develops flexible parametric methods based on splines to estimate the hazard function hi t that describes the rate of disease onset among persons still susceptible at time t the observed data derive from a random sample yi i 1 n n of disease onset times with survivorship function s t py t the yi are right censored by independent random variables ci drawn from h c p c c one observes xi min yi ci and 3i i y ci the indicator for the event that yi is censored the hazard function

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

projection estimation in multiple regression with application to functional anova models a general theory on rates of convergence of the least squares projection estimate in multiple regression is developed the theory is applied to the functional anova model where the multivariate regression function is modeled as a specified sum of a constant term main effects functions of one variable and selected interaction terms functions of two or more variables the least squares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed anova structure of the regression function the linear spaces that serve as building blocks can be any of the ones commonly used in practice polynomials trigonometric polynomials splines wavelets and finite elements the rate of convergence result that is obtained reinforces the intuition that low order anova modeling can achieve dimension reduction and thus overcome the curse of dimensionality moreover the components of the projection estimate in an appropriately defined anova decomposition provide consistent estimates of the corresponding components of the regression function when the regression function does not satisfy the assumed anova form the projection estimate converges to its best approximation of that form

measuring synchronization in coupled model systems a comparison of different approaches the investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation for experimental data the underlying dynamics are usually not completely known therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis in this study we use three different coupled model systems to create a controlled setting for a comparison of six different measures of synchronization all measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications although certain dynamical features of a system under investigation e g power spectra dimension may render certain measures more suitable than others c 2006 elsevier b v all rights reserved

an adaptive orthogonal search algorithm for model subset selection and non linear system identification a new adaptive orthogonal search aos algorithm is proposed for model subset selection and non linear system identification model structure detection is a key step in any system identification problem this consists of selecting significant model terms from a redundant dictionary of candidate model terms and determining the model complexity model length or model size the final objective is to produce a parsimonious model that can well capture the inherent dynamics of the underlying system in the new aos algorithm a modified generalized cross validation criterion called the adjustable prediction error sum of squares apress is introduced and incorporated into a forward orthogonal search procedure the main advantage of the new aos algorithm is that the mechanism is simple and the implementation is direct and easy and more importantly it can produce efficient model subsets for most non linear identification problems

persymmetric jacobi matrices isospectral deformations and orthogonal polynomials abstract persymmetric jacobi matrices are invariant under reflection with respect to the anti diagonal the associated orthogonal polynomials have distinctive properties that are discussed they are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice this is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric jacobi matrices from spectral points isospectral deformations of such matrices are also considered expressions for the associated polynomials and their weights are obtained in terms of the undeformed entities

monotone spline transformations for dimension reduction consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann p matrix a monotone transformation of the values in each column represented as a linear combination of integrated basis splines is assumed determined by a linear combination of a new set of values characterizing each row object two different models are used one an eckart young decomposition model and the other a multivariate normal model examples for artificial and real data are presented the results indicate that both methods are helpful in choosing dimensionality and that the eckart young model is also helpful in displaying the relationships among the objects and the variables also results suggest that the resulting transformations are themselves illuminating

extended hazard regression for censored survival data with covariates a spline approximation for the baseline hazard function a regression model for censored survival data with covariates is introduced and termed extended hazard regression ehr ehr includes the proportional hazards ph and the accelerated failure time aft models as special cases by approximating the baseline hazard function with quadratic splines we develop a maximum likelihood estimation procedure to provide a simultaneous estimate of both the hazard function and the regression coefficients the aft and ph assumptions can then be compared by likelihood ratio tests within the more general ehr model several examples based on artificial and real data are discussed

regression analysis of censored survival data with the generalized f family an alternative to the proportional hazards model we show that the generalized f family is a useful tool for regression analysis with censored survival data we discuss maximum likelihood estimation and give asymptotic procedures for calculating confidence intervals tests of significance for the parameters life expectancy quantiles and survival rates survival data on 704 ovarian carcinoma patients serve to demonstrate the utility of the model

a variable span smoother abstract a variable span smoother based on linear fits is described local cross validation is used to estimate the optimal span as a function of abscissa value computationally efficient algorithms making use of updating formulas are presented

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

flexible regression models with cubic splines we describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates this simple method can help prevent the problems that result from inappropriate linearity assumptions we compare restricted cubic spline regression to non parametric procedures for characterizing the relationship between age and survival in the stanford heart transplant data we also provide an illustrative example in cancer therapeutics

how many variables should be entered in a regression equation abstract the optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points as the number of regressors grows large the sp criterion provides an asymptotically optimal rule for the number of variables to enter

nonuniform state space reconstruction and coupling detection we investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past current and future states the embedding scheme can be adapted for different purposes such as mixed modeling cross prediction and granger causality in particular we apply this method in order to detect and evaluate information transfer in coupled systems as a practical application we investigate in records of scalp epileptic eeg the information flow across brain areas

separation theorems for the zeros of certain hypergeometric polynomials we study interlacing properties of the zeros of two contiguous 2f1 hypergeometric polynomials we use the connection between hypergeometric 2f1 and jacobi polynomials as well as a monotonicity property of zeros of orthogonal polynomials due to markoff to prove that the zeros of contiguous hypergeometric polynomials separate each other we also discuss interlacing results for the zeros of 2f1 and those of the polynomial obtained by shifting one of the parameters of 2f1 by t where 0 t 1

a q generalization of the para racah polynomials abstract new bispectral orthogonal polynomials are obtained from an unconventional truncation of the askey wilson polynomials in the limit q 1 they reduce to the para racah polynomials which are orthogonal with respect to a quadratic bi lattice the three term recurrence relation and q difference equation are obtained through limits of those of the askey wilson polynomials an explicit expression in terms of hypergeometric series and the orthogonality relation are provided a q generalization of the para krawtchouk polynomials is obtained as a special case connections with the q racah and dual hahn polynomials are also presented

link based models for survival data with interval and continuous time censoring we present a flexible family of link based regression models with time independent baseline covariates particular choices of the link function yield a proportional hazard or a proportional odds model among others the background hazard function in the model is estimated using b splines this yields a full likelihood under both continuous time and random interval censoring residuals for influence and goodness of fit are described two examples are shown one in continuous time and the other under interval censoring

a characterization of classical and semiclassical orthogonal polynomials from their dual polynomials we study properties of the dual orthogonal polynomials op introduced by de boor and saff and present a new characterization of the classical and semiclassical op

construction of acyclic matrices from spectral data abstract given real numbers 1 n 1 n 1 with 1 2 2 n 1 n an integer i with 1 i n and a tree with n vertices we prove the existence of a hermitian acyclic matrix a x rs with eigenvalues 1 n and x rs 0 whenever r s and r s is not an edge of and such that the principal submatrix of a obtained from a by deleting the i th row and column has eigenvalues 1 n 1

finite sequences of orthogonal polynomials connected by a jacobi matrix abstract if p0 pn is an orthogonal sequence with pj a monic polynomial of exact degree j all j then the sequence q0 qn of monic polynomials defined by the requirement that pj cjqn j 1pn 1 at the zeros of pn with cj suitable constants is also orthogonal with respect to a possibly differently weighted scalar product these two sequences can be shown to be the upper left respectively lower right principal minors of the matrix t j with j a suitable tridiagonal matrix the sequence q occurs in the characterization of the discrete least squares approximation to f from span p0 pj in terms of the n j 1 order divided differences of f p n 1 at the nodes i e the zeros of pn a complete characterization of the pair u v of zero sets of pj 1 and qn j is given and an application is made to the problem of recovering a jacobi matrix from such data

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

an extension of the proportional hazards model for grouped data summary a family of transformations for probabilities is considered for the analysis of grouped survival data additive and multiplicative models for the hazard function are compared a method is suggested for estimating the scale on which an additive representation of the hazard in terms of explanatory variables is appropriate computations are performed using glim a test for departures from the grouped proportional hazards model is introduced an example is analyzed by the methods put forward in the analysis of survival data the effects of explanatory variables on failure time t are of great interest these effects are perhaps most conveniently investigated by establishing simple models for the hazard function in terms of explanatory variables we focus attention on additive and multiplicative models for the hazard the approach taken is to analyze sequences of contingency tables constructed from the data this follows mantel 1966 though the way in which we determine the sequences parallels work by holford 1976 certain similarities to problems involving binary data are exploited in particular a family of transformations for probabilities is defined which includes both additive and multiplicative models as special cases generalized linear models nelder and wedderburn 1972 associated with the transformations may be tried on the data and their fits compared in 3 a test for departures from the grouped multiplicative model in the direction of the grouped additive model is proposed and the forms of two simple estimates of the survivor function are given the methods are applied to an example in 4

regression splines in the cox model with application to covariate effects in liver disease abstract the cox proportional hazards model restricts the log hazard ratio to be linear in the covariates a smooth nonlinear covariate effect may go undetected in this model but can be well approximated by a spline function a survival model based on data from a clinical trial of primary biliary cirrhosis is developed using regression splines and the resulting log hazard ratio estimates are compared with those from nonparametric methods we remove the linear restriction on the log hazard ratio by transforming a continuous covariate into a vector of fixed knot basis splines b splines b splines are known to produce better conditioned systems of equations than the truncated power basis when used as interpolants and show similar behavior when fitting proportional hazards models we describe the procedures for and the issues arising in the estimation and the testing of the b spline coefficients although inference is not well developed for some nonparametric methods that estimate covariate effects the

regression analysis with randomly right censored data this paper proposes a new estimator of the parameter vector in a linear regression model when the observations are randomly censored on the right and when the error distribution is unknown this estimator is explicitly defined and easily computable the paper contains sufficient conditions under which this estimator is mean square consistent and asymptotically normal a numerical example is given

variations around classical orthogonal polynomials connected problems abstract we survey and prove the principal characterizations of the classical orthogonal polynomials with the method of the dual sequence two new characterizations are given connected problems are discussed

a constrained spline estimator of a hazard function a constrained quadratic spline is proposed as an estimator of the hazard function of a random variable a maximum penalized likelihood procedure is used to fit the estimator to a sample of psychological response times the results of a small simulation study suggest that with a sample size of 500 the procedure may provide a reasonably precise estimate of the shape of a hazard function

cross validation the jackknife and the bootstrap excess error estimation in forward logistic regression abstract given a prediction rule based on a set of patients what is the probability of incorrectly predicting the outcome of a new patient call this probability the true error an optimistic estimate is the apparent error or the proportion of incorrect predictions on the original set of patients and it is the goal of this article to study estimates of the excess error or the difference between the true and apparent errors i consider three estimates of the excess error cross validation the jackknife and the bootstrap using simulations and real data the three estimates for a specific prediction rule are compared when the prediction rule is allowed to be complicated overfitting becomes a real danger and excess error estimation becomes important the prediction rule chosen here is moderately complicated involving a variable selection procedure based on forward logistic regression

an approach for characterizing coupling in dynamical systems abstract the study of coupling in dynamical systems dates back to christian hyugens who in 1665 discovered that pendulum clocks with the same length pendulum synchronize when they are near to each other in that case the observed synchronous motion was out of phase in this paper we propose a new approach for measuring the degree of coupling and synchronization of a dynamical system consisting of interacting subsystems the measure is based on quantifying the active degrees of freedom e g correlation dimension of the coupled system and the constituent subsystems the time delay embedding scheme is extended to coupled systems and used for attractor reconstruction of the coupled dynamical system we use the coupled lorenz rossler and henon model systems with a coupling strength variable for evaluation of the proposed approach results show that we can measure the active degrees of freedom of the coupled dynamical systems and can quantify and distinguish the degree of synchronization or coupling in each of the dynamical systems studied furthermore using this approach the direction of coupling can be determined

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

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

understanding cox s regression model a martingale approach abstract an informal discussion is given of how martingale techniques can be used to extend cox s regression model and to derive its large sample properties

interlacing properties of the zeros of the orthogonal polynomials and approximation of the hilbert transform abstract in this paper starting from interlacing properties of the zeros of the orthogonal polynomials the authors propose a new method to approximate the finite hilbert transform for this method they give error estimates in uniform norm

zeros of linear combinations of laguerre polynomials from different sequences we study interlacing properties of the zeros of two types of linear combinations of laguerre polynomials with different parameters namely r n l n a al n a and s n l n a bl n 1 a proofs and numerical counterexamples are given in situations where the zeros of r n and s n respectively interlace or do not in general with the zeros of l k a l k a k n or n 1 the results we prove hold for continuous as well as integral shifts of the parameter a

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

the uniqueness of an orthogonality measure given two finite sets of real points z n j j 1 n and z n 1 j j 1 n 1 satisfying the interlacing property z n 1 j

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

a multilevel excess hazard model to estimate net survival on hierarchical data allowing for non linear and non proportional effects of covariates the excess hazard regression model is an approach developed for the analysis of cancer registry data to estimate net survival that is the survival of cancer patients that would be observed if cancer was the only cause of death cancer registry data typically possess a hierarchical structure individuals from the same geographical unit share common characteristics such as proximity to a large hospital that may influence access to and quality of health care so that their survival times might be correlated as a consequence correct statistical inference regarding the estimation of net survival and the effect of covariates should take this hierarchical structure into account it becomes particularly important as many studies in cancer epidemiology aim at studying the effect on the excess mortality hazard of variables such as deprivation indexes often available only at the ecological level rather than at the individual level we developed here an approach to fit a flexible excess hazard model including a random effect to describe the unobserved heterogeneity existing between different clusters of individuals and with the possibility to estimate non linear and time dependent effects of covariates we demonstrated the overall good performance of the proposed approach in a simulation study that assessed the impact on parameter estimates of the number of clusters their size and their level of unbalance we then used this multilevel model to describe the effect of a deprivation index defined at the geographical level on the excess mortality hazard of patients diagnosed with cancer of the oral cavity copyright 2016 john wiley sons ltd

semiparametric efficient estimation for a class of generalized proportional odds cure models we present a mixture cure model with the survival time of the uncured group coming from a class of linear transformation models which is an extension of the proportional odds model this class of model first proposed by dabrowska and doksum 1988 which we term generalized proportional odds model is well suited for the mixture cure model setting due to a clear separation between long term and short term effects a standard expectation maximization algorithm can be employed to locate the nonparametric maximum likelihood estimators which are shown to be consistent and semiparametric efficient however there are difficulties in the m step due to the nonparametric component we overcome these difficulties by proposing two different algorithms the first is to employ an majorize minimize mm algorithm in the m step instead of the usual newton raphson method and the other is based on an alternative form to express the model as a proportional hazards frailty model the two new algorithms are compared

comparing smoothing techniques in cox models for exposure response relationships to allow for non linear exposure response relationships we applied flexible non parametric smoothing techniques to models of time to lung cancer mortality in two occupational cohorts with skewed exposure distributions we focused on three different smoothing techniques in cox models penalized splines restricted cubic splines and fractional polynomials we compared standard software implementations of these three methods based on their visual representation and criterion for model selection we propose a measure of the difference between a pair of curves based on the area between them standardized by the average of the areas under the pair of curves to capture the variation in the difference over the range of exposure the area between curves was also calculated at percentiles of exposure and expressed as a percentage of the total difference the dose response curves from the three methods were similar in both studies over the denser portion of the exposure range with the difference between curves up to the 50th percentile less than 1 per cent of the total difference a comparison of inverse variance weighted areas applied to the data set with a more skewed exposure distribution allowed us to estimate area differences with more precision by reducing the proportion attributed to the upper 1 per cent tail region overall the penalized spline and the restricted cubic spline were closer to each other than either was to the fractional polynomial

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

duality of orthogonal polynomials on a finite set we prove a certain duality relation for orthogonal polynomials defined on a finite set the result is used in a direct proof of the equivalence of two different ways using particles or holes of computing the correlation functions of a discrete orthogonal polynomial ensemble

on toda lattices and orthogonal polynomials first we derive a simple connection between toda and langmuir lattices and give a characterization of toda lattices with the help of stieltjes functions then it is shown how to generate by orthogonal polynomials in an elementary way periodic and almost periodic toda lattices the particles of the toda lattice are not even restricted as usual to move on the real line they may also move in the complex plane with the help of this result for special cases explicit solutions are obtained in terms of elliptic functions

accelerated hazards model based on parametric families generalized with bernstein polynomials a transformed bernstein polynomial that is centered at standard parametric families such as weibull or log logistic is proposed for use in the accelerated hazards model this class provides a convenient way towards creating a bayesian nonparametric prior for smooth densities blending the merits of parametric and nonparametric methods that is amenable to standard estimation approaches for example optimization methods in sas or r can yield the posterior mode and asymptotic covariance matrix this novel nonparametric prior is employed in the accelerated hazards model which is further generalized to time dependent covariates the proposed approach fares considerably better than previous approaches in simulations data on the effectiveness of biodegradable carmustine polymers on recurrent brain malignant gliomas is investigated

nonparametric analysis of an accelerated failure time model summary survival distributions can be characterized by and compared through their hazard functions tests using a proportional hazards model have good power if the two hazards do not cross but without time dependent covariates can have low power if they do the method contained herein is designed to provide a complement to the proportional hazards model differences in survival distributions are parameterized by a scale change and the log rank statistic is used to generate an estimate of the change and a confidence interval based test this approach is fully efficient for the weibull family with no censoring in general it compares favourably with the proportional hazards approach although neither method dominates

the decomposition of time varying hazard into phases each incorporating a separate stream of concomitant information abstract the hazard function of time related events such as death or reoperation following heart valve replacement often is time varying in a structured fashion as is the influence of risk factors associated with the events a completely parametric system is presented for the decomposition of time varying patterns of risk into additive overlapping phases descriptively labeled as early constant hazard and late each phase is shaped by a different generic function of time constituting a family of nested equations and is scaled by a separate logit linear or log linear function of concomitant information model building uses maximum likelihood estimation the resulting parametric equations permit hazard function survivorship function and probability estimates and their confidence limits to be portrayed and adjusted for concomitant information these provide a comprehensive analysis of time related events from which inferences may be drawn to improve for example the management of patients with valva

survival estimation using splines a nonparametric maximum likelihood procedure is given for estimating the survivor function from right censored data it approximates the hazard rate by a simple function such as a spline with different approximations yielding different estimators a special case is that proposed by nelson 1969 journal of quality technology 1 27 52 and altshuler 1970 mathematical biosciences 6 1 11 the estimators are uniformly consistent and have the same asymptotic weak convergence properties as the kaplan meier 1958 journal of the american statistical association 53 457 481 estimator however in small and in heavily censored samples the simplest spline estimators have uniformly smaller mean squared error than do the kaplan meier and nelson altshuler estimators the procedure is extended to estimate the baseline hazard rate and regression coefficients in the cox 1972 journal of the royal statistical society series b 34 187 220 proportional hazards model and is illustrated using experimental carcinogenesis data

on neighbouring matrices with quadratic elementary divisors this paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrixa and relates them to those given by kakan it shows that all results of this kind may give severe overestimates of the distance to the nearest defective matrix and that matrices with elementary divisors of degrees higher than two are often at a much greater distance than those with quadratic divisors

parametric or nonparametric a parametricness index for model selection in model selection literature two classes of criteria perform well asymptotically in different situations bayesian information criterion bic as a representative is consistent in selection when the true model is finite dimensional parametric scenario akaike s information criterion aic performs well in an asymptotic efficiency when the true model is infinite dimensional nonparametric scenario but there is little work that addresses if it is possible and how to detect the situation that a specific model selection problem is in in this work we differentiate the two scenarios theoretically under some conditions we develop a measure parametricness index pi to assess whether a model selected by a potentially consistent procedure can be practically treated as the true model which also hints on aic or bic is better suited for the data for the goal of estimating the regression function a consequence is that by switching between aic and bic based on the pi the resulting regression estimator is simultaneously asymptotically efficient for both parametric and nonparametric scenarios in addition we systematically investigate the behaviors of pi in simulation and real data and show its usefulness

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

visualization of coupling in time series by order recurrence plots we introduce a method to visualize dependencies between two time series by applying the concept of cross recurrence plots to the local ordinal structure we derive a measure of the coupling strength which is robust against observational noise nonlinear distortion of the amplitude and low frequency trends connections to the instantaneous phase and the determination of phase coupling of two coupled rossler systems in the standard and funnel regimes are shown an application to electroencephalogram data demonstrates that the method is robust with respect to artifacts

spatial extended hazard model with application to prostate cancer survival summary this article develops a bayesian semiparametric approach to the extended hazard model with generalization to high dimensional spatially grouped data county level spatial correlation is accommodated marginally through the normal transformation model of li and lin 2006 journal of the american statistical association 101 591 603 using a correlation structure implied by an intrinsic conditionally autoregressive prior efficient markov chain monte carlo algorithms are developed especially applicable to fitting very large highly censored areal survival data sets per variable tests for proportional hazards accelerated failure time and accelerated hazards are efficiently carried out with and without spatial correlation through bayes factors the resulting reduced interpretable spatial models can fit significantly better than a standard additive cox model with spatial frailties

linear combinations of orthogonal polynomials generating positive quadrature formulas let pk x x k e n0 be the polynomials orthogonal on 1 1 with respect to the positive measure da we give sufficient conditions on the real numbers p j 0 m such that the linear combination of orthogonal polynomials yflo jpn j has n simple zeros in 1 1 1 and that the interpolatory quadrature formula whose nodes are the zeros of yfj osjpn j has positive weights

interlacing of zeros of shifted sequences of one parameter orthogonal polynomials we study the interlacing property of zeros of laguerre polynomials of adjacent degree where the free parameters differ by an integer and of the same degree where the free parameter is shifted continuously similar interlacing results are proven for the positive zeros of gegenbauer polynomials

on the behavior of certain maximum likelihood estimators from large randomly censored samples abstract in the case the survival distribution of a life length x is of the form ln 1 fx x q x for x 0 where the hazard function q is known but or or both are unknown the asymptotic distribution of the maximum likelihood estimators of the shape and scale parameters is found for large randomly censored samples the covariance matrix of is expressed in terms of the distribution of the random variable that can be observed namely y min x t here t is the random censoring time independent of x sufficient conditions are found such as t having only finite support to ensure that the estimates are strongly consistent and asymptotically normal application is made to decreasing hazard rate distributions

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

extended lagrange interpolation in weighted uniform norm the author studies the uniform convergence of extended lagrange interpolation processes based on the zeros of generalized laguerre polynomials

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

the restricted cubic spline as baseline hazard in the proportional hazards model with step function time dependent covariables we incorporate a cubic spline function where the tails are linearly constrained as the baseline hazard into the proportional hazards model we show estimation of covariable coefficients and survival probabilities with this model to be as efficient statistically as with the cox proportional hazards model when covariables are fixed examples show that the inclusion of time dependent covariables defined as step functions into the restricted cubic spline proportional hazards model reduces computation time by a factor of 213 over the cox model advantages of the spline model also include flexibility of the hazard smooth survival curves and confidence limits for the survival and hazard estimates when there are time dependent covariables present

on quantile regression in reproducing kernel hilbert spaces with the data sparsity constraint for spline regressions it is well known that the choice of knots is crucial for the performance of the estimator as a general learning framework covering the smoothing splines learning in a reproducing kernel hilbert space rkhs has a similar issue however the selection of training data points for kernel functions in the rkhs representation has not been carefully studied in the literature in this paper we study quantile regression as an example of learning in a rkhs in this case the regular squared norm penalty does not perform training data selection we propose a data sparsity constraint that imposes thresholding on the kernel function coefficients to achieve a sparse kernel function representation we demonstrate that the proposed data sparsity method can have competitive prediction performance for certain situations and have comparable performance in other cases compared to that of the traditional squared norm penalty therefore the data sparsity method can serve as a competitive alternative to the squared norm penalty method some theoretical properties of our proposed method using the data sparsity constraint are obtained both simulated and real data sets are used to demonstrate the usefulness of our data sparsity constraint