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

bayesian inferences on umbrella orderings in regression applications with categorical predictors interest often focuses on comparing the null hypothesis of homogeneity to an ordered alternative this article proposes a bayesian approach for addressing this problem in the setting of normal linear and probit regression models the regression coefficients are assigned a conditionally conjugate prior density consisting of mixtures of point masses at 0 and truncated normal densities with a possibly unknown changepoint parameter included to accommodate umbrella ordering two strategies of prior elicitation are considered 1 a bayesian bonferroni approach in which the probability of the global null hypothesis is specified and local hypotheses are considered independent and 2 an approach which treats these probabilities as random a single gibbs sampling chain can be used to obtain posterior probabilities for the different hypotheses and to estimate regression coefficients and predictive quantities either by model averaging or under the preferred hypothesis the methods are applied to data from a carcinogenesis study

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

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

a general class of parametric models for recurrent event data the general class of models proposed by pena and hollander for recurrent event data is considered under a fully parametric specification of the baseline hazard rate function and under the two cases where the model does and does not incorporate frailty components estimators of model parameters are presented and their finite and asymptotic properties are ascertained for the asymptotic properties the results of borgan concerning maximum likelihood estimators in counting process models are used to obtain weak convergence to gaussian distributions of estimators however the required regularity conditions are reformulated into conditions involving gap times which make it more feasible to obtain explicit theoretical expressions of asymptotic covariances the procedures are applied to fit the general class of models with a parametric baseline hazard rate function to a dataset on hydraulic subsystems of load haul dump machines in mining

generalized additive and index models with shape constraints summary we study generalized additive models with shape restrictions e g monotonicity convexity and concavity imposed on each component of the additive prediction function we show that this framework facilitates a non parametric estimator of each additive component obtained by maximizing the likelihood the procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals more generally our methodology can be applied to generalized additive index models here again the procedure can be justified on theoretical grounds and like the original algorithm has highly competitive finite sample performance practical utility is illustrated through the use of these methods in the analysis of two real data sets our algorithms are publicly available in the r package scar short for shape constrained additive regression

on the estimation of an instantaneous transformation for time series summary we modify ramsay s algorithm for estimating monotonic transformations in regression and extend it to autoregression where strict monotonicity is an essential requirement compared with other methods our method can capture some characteristics that are pertinent to the time series and is much easier to implement an order selection method is introduced and developed some real data sets are analysed

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

unified approach to testing functional hypotheses in semiparametric contexts abstract we suggest a new method with very wide applicability for testing semiparametric hypotheses about functions such as regression means and probability densities the technique is based on characterising hypotheses in terms of functionals which can be estimated root n consistently and constructing test statistics in terms of estimators of the functionals since the tests are semiparametric it is appropriate to assess them on the basis of their ability to detect departures of size n 1 2 from the null hypothesis we show that they do indeed have this property unlike tests constructed in a nonparametric setting their power does not depend critically on choice of a bandwidth and in particular smoothing parameter selection is not an issue that has to be addressed by users of the tests bootstrap methods are suggested for calibrating the tests in a regression setting applications include tests of specification such as partial linear and index models against nonparametric or semiparametric alternatives and tests of monotonicity concavity separability equality of regression functions and base independence of equivalence scales in a density setting they include tests of radial symmetry and stochastic dominance

an elementary approach to weak convergence for quantile processes with applications to censored survival data abstract let be a continuously differentiable function with positive derivative and let n be a sequence of right continuous increasing processes we show that if n 1 2 n w where w is continuous then n 1 2 1 n 1 w 1 1 this result is applied to classical processes such as the empirical distribution function the kaplan meier estimator and some other situations we also prove an analogous result for the bootstrapped version of n 1 2 n and show how this allows one to obtain confidence bands for the quantile function 1 based on the bootstrap several examples are given

a fast scaling algorithm for minimizing separable convex functions subject to chain constraints we consider the problem of minimizing j nc j x j subject to the following chain constraints x 1 x 2 x 3 x n where c j x j is a convex function of x j for each j n 1 2 n this problem is a generalization of theisotonic regression problems with complete order an important class of problems in regression analysis that has been examined extensively in the literature we refer to this problem as thegeneralized isotonic regression problem in this paper we focus on developing a fast scaling algorithm to obtain an integer solution of the generalized isotonic regression problem letu denote the difference between an upper bound on an optimal value of x n and a lower bound on an optimal value of x 1 under the assumption that the evaluation of any function c j x j takeso 1 time we show that the generalized isotonic regression problem can be solved ino n logu time this improves by a factor of n the previous best running time ofo n 2 logu to solve the same problem in addition when our algorithm is specialized to theisotonic median regression problem where c j x j c j x j a ji for specified values ofc js anda js the algorithm obtains a real valued optimal solution ino n logn time this time bound matches the best available time bound to solve the isotonic median regression problem but our algorithm uses simpler data structures and may be easier to implement

calibration using constrained smoothing with applications to mass spectrometry data summary linear regressions are commonly used to calibrate the signal measurements in proteomic analysis by mass spectrometry however with or without a monotone e g log transformation data from such functional proteomic experiments are not necessarily linear or even monotone functions of protein or peptide concentration except over a very restricted range a computationally efficient spline procedure improves upon linear regression however mass spectrometry data are not necessarily homoscedastic more often the variation of measured concentrations increases disproportionately near the boundaries of the instruments measurement capability dynamic range that is the upper and lower limits of quantitation these calibration difficulties exist with other applications of mass spectrometry as well as with other broad scale calibrations therefore the method proposed here uses a functional data approach to define the calibration curve and also the limits of quantitation under the two assumptions i that the variance is a bounded convex function of concentration and ii that the calibration curve itself is monotone at least between the limits of quantitation but not necessarily outside these limits within this paradigm the limit of detection where the signal is definitely present but not measurable with any accuracy is also defined an iterative approach draws on existing smoothing methods to account simultaneously for both restrictions and is shown to achieve the global optimal convergence rate under weak conditions this approach can also be implemented when convexity is replaced by other bounded restrictions examples from addona et al 2009 nature biotechnology 27 663 641 both motivate and illustrate the effectiveness of this functional data methodology when compared with the simpler linear regressions and spline techniques

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

confidence interval procedures for the weibull process with applications to reliability growth the weibull process a nonhomogeneous poisson process with intensity r t t 1 is considered as a stochastic model for the duane 1964 reliability growth postulate under this model the mean time between failure mtbf for the system at time t is given by m t r t 1 small sample and asymptotic confidence intervals on m t are discussed for failure and time truncated testing tabled values to compute the confidence intervals and numercial examples illustrating these procedures are presented

tests for an increasing trend in the intensity of a poisson process a power study abstract this article concerns a comparison of several tests for testing the hypothesis of a constant intensity against the alternative of an increasing intensity function in a nonhomogeneous poisson process nhpp the study includes the well known laplace test statistic the most powerful test for the shape parameter in a poisson process with weibull intensity the likelihood ratio test against arbitrary nhpp alternatives two nonparametric tests for trends based on kendall s tau and spearman s rho and a test based on an f statistic the powers of the tests are determined by monte carlo simulation against alternatives that are increasing at an exponential rate a power rate weibull intensity and a logarithmic rate alternatives that are step functions with one jump are also considered in a few cases the exact powers are also obtained analytically

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

discontinuous versus smooth regression given measurements x i y i i 1 n we discuss methods to assess whether an underlying regression function is smooth continuous or differentiable or whether it has discontinuities the variance of the measurements is assumed to be unknown and is estimated simultaneously by regressing squared differences of the data formed with various span sizes on the span size itself we obtain an asymptotic linear model with dependent errors the parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements both parameters can be consistently estimated with mean squared error rates of convergence of n 2 3 for the sum of squared jump sizes and n 1 for the error variance we derive the asymptotic constants of the mean squared error mse and discuss the dependence of mse on the maximum span size l the test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0 the asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for u statistics we motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps

non parametric estimation of the odds ratios for continuous exposures using generalized additive models with an unknown link function the generalized additive model gam is a powerful and widely used tool that allows researchers to fit non parametrically the effect of continuous predictors on a transformation of the mean response variable such a transformation is given by a so called link function and in gams this link function is assumed to be known nevertheless if an incorrect choice is made for the link the resulting gam is misspecified and the results obtained may be misleading in this paper we propose a modified version of the local scoring algorithm that allows for the non parametric estimation of the link function by using local linear kernel smoothers to better understand the effect that each covariate produces on the outcome results are expressed in terms of the non parametric odds ratio or curves bootstrap techniques were used to correct the bias in the or estimation and to construct point wise confidence intervals a simulation study was carried out to assess the behaviour of the resulting estimates the proposed methodology was illustrated using data from the aids register of galicia nw spain with a view to assessing the effect of the cd4 lymphocyte count on the probability of being aids diagnosed via tuberculosis tb this application shows how the link s flexibility makes it possible to obtain or curve estimates that are less sensitive to the presence of outliers and unusual values that are often present in the extremes of the covariate distributions

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

the statistical analysis of failure time data preface 1 introduction 1 1 failure time data 1 2 failure time distributions 1 3 time origins censoring and truncation 1 4 estimation of the survivor function 1 5 comparison of survival curves 1 6 generalizations to accommodate delayed entry 1 7 counting process notation bibliographic notes exercises and complements 2 failure time models 2 1 introduction 2 2 some continuous parametric failure time models 2 3 regression models 2 4 discrete failure time models bibliographic notes exercises and complements 3 inference in parametric models and related topics 3 1 introduction 3 2 censoring mechanisms 3 3 censored samples from an exponential distribution 3 4 large sample likelihood theory 3 5 exponential regression 3 6 estimation in log linear regression models 3 7 illustrations in more complex data sets 3 8 discrimination among parametric models 3 9 inference with interval censoring 3 10 discussion bibliographic notes exercises and complements 4 relative risk cox regression models 4 1 introduction 4 2 estimation of beta 4 3 estimation of the baseline hazard or survivor function 4 4 inclusion of strata 4 5 illustrations 4 6 counting process formulas 4 7 related topics on the cox model 4 8 sampling from discrete models bibliographic notes exercises and complements 5 counting processes and asymptotic theory 5 1 introduction 5 2 counting processes and intensity functions 5 3 martingales 5 4 vector valued martingales 5 5 martingale central limit theorem 5 6 asymptotics associated with chapter 1 5 7 asymptotic results for the cox model 5 8 asymptotic results for parametric models 5 9 efficiency of the cox model estimator 5 10 partial likelihood filtration bibliographic notes exercises and complements 6 likelihood construction and further results 6 1 introduction 6 2 likelihood construction in parametric models 6 3 time dependent covariates and further remarks on likelihood construction 6 4 time dependence in the relative risk model 6 5 nonnested conditioning events 6 6 residuals and model checking for the cox model bibliographic notes exercises and complements 7 rank regression and the accelerated failure time model 7 1 introduction 7 2 linear rank tests 7 3 development and properties of linear rank tests 7 4 estimation in the accelerated failure time model 7 5 some related regression models bibliographic notes exercises and complements 8 competing risks and multistate models 8 1 introduction 8 2 competing risks 8 3 life history processes bibliographic notes exercises and complements 9 modeling and analysis of recurrent event data 9 1 introduction 9 2 intensity processes for recurrent events 9 3 overall intensity process modeling and estimation 9 4 mean process modeling and estimation 9 5 conditioning on aspects of the counting process history bibliographic notes exercises and complements 10 analysis of correlated failure time data 10 1 introduction 10 2 regression models for correlated failure time data 10 3 representation and estimation of the bivariate survivor function 10 4 pairwise dependency estimation 10 5 illustration australian twin data 10 6 approaches to nonparametric estimation of the bivariate survivor function 10 7 survivor function estimation in higher dimensions bibliographic notes exercises and complements 11 additional failure time data topics 11 1 introduction 11 2 stratified bivariate failure time analysis 11 3 fixed study period survival studies 11 4 cohort sampling and case control studies 11 5 missing covariate data 11 6 mismeasured covariate data 11 7 sequential testing with failure time endpoints 11 8 bayesian analysis of the proportional hazards model 11 9 some analyses of a particular data set bibliographic notes exercises and complements glossary of notation appendix a some sets of data appendix b supporting technical material bibliography author index subject index

statistical analysis of repeated events forming renewal processes for each of several individuals a sequence of repeated events forming a renewal process is observed up to some censoring time the object is to estimate the average interevent time over the population of individuals as well as the variation of interevent times within and between individuals medical motivation comes from gastroenterology and concerns the occurrence of certain cyclic movements in the small bowel during the fasting state two statistical models are considered one is the standard variance component model adapted to censored data and the other is a recent intensity based model with a random proportionality factor representing interindividual variation these models are applied to the motility data and their advantages are discussed the intensity based model allows simple empirical bayes estimation of the expected interevent times for an individual in the presence of censoring

level crossing probabilities for multipath acoustic processes with bimodal spectra a general method of evaluating upcrossing probabilities for a class of random processes consisting of two narrow band signals is presented one of the two significant frequencies of the corresponding bimodal spectra is assumed to be dominant the method approximates the maxima of these processes by the corresponding values of the envelope processes it is also assumed that the discrete processes of the maxima are markov the results have several applications two prominent examples are detection problems of multipath partially saturated processes in underwater acoustics and the problem of the structural reliability of marine diesel engine shafting systems

direct regression models for longitudinal rates of change comparing rates of growth or rates of change across covariate defined subgroups is a primary objective for many longitudinal studies in the special case of a linear trend over time the interaction between a covariate and time will characterize differences in longitudinal rates of change however in the presence of a non linear longitudinal trajectory the standard mean regression approach does not permit parsimonious description or inference regarding differences in rates of change therefore we propose regression methodology for longitudinal data that allows a direct structured comparison of rates across subgroups even in the presence of a non linear trend over time our basic longitudinal rate regression method assumes a proportional difference across covariate groups in the rate of change across time but this assumption can be relaxed rates are compared relative to a generally specified time trend for which we discuss both parametric and non parametric estimating approaches we develop mixed model longitudinal methodology that explicitly characterizes subject to subject variation in rates as well as a marginal estimating equation based method in addition we detail a score test to detect violations of the proportionality assumption and we allow time varying rate effects as a natural generalization simulation results demonstrate potential gains in power for the longitudinal rate regression model relative to a linear mixed effects model in the presence of a non linear trend in time we apply our method to a study of growth among infants born to hiv infected mothers and conclude with a discussion of possible extensions for our methods

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

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

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

some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates abstract recurrent event time data are common in medical research examples include infections in aids patients and seizures in epilepsy patients in this context as well as in the more usual context of a single failure time variable time dependent covariates are frequently of interest we suggest some rate functions that might be displayed when analyzing recurrent failure time data or when the effect of a categorical time dependent covariate is of interest estimators of these functions are provided along with two sample test statistics a new approach to regression modeling of these data is suggested and contrasted with existing methods our methods do not require that an explicit model be formulated for the probabilistic association between failure times within an individual this is in line with the currently popular generalized estimating equation approach to longitudinal data if the nature of such associations is known or is of particular interest then alternative methods may be appropriate

maximally selected chi square statistics two samples can be compared by selecting a cut point and then forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample when the cut point is selected so as to maximize the standard chi square statistic the chi square percentile points are inappropriate actual significance levels are computed for large samples and correct percentile poillts are tabulated

tables for testing ordered alternatives in an analysis of variance without replications summary given a series of k normal populations with a common but unknown variance and supposing that an observation is drawn from each one of them then a likelihood ratio test of homogeneity of the means against the simple order alternative is possible when the significance level oc is less than 1 k for these possible combinations of a and k tables for testing the hypothesis are provided where oc 0 10 0 05 0 025 0 01 0 005 0 0025

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

new models on the ocean acoustic detection process the basic problem of the ocean acoustic detection process is formulated analytically under the assumption of fully developed saturated phase random multipath acoustic fluctuations detection is defined as occurring whenever p the root mcan square pressure at the receiver exceeds a specified threshold level p two models one exact and one approximate are developed for obtaining the probability density functions of the time between two successive detections and of the time p is above p0 holding time the two models are compared with one another and with the extensively used 2 r model one of the reasons that the latter model has a limited success in practice is the inability to estimate the appropriate value for 2 a parameter which is determined empirically in this paper we have derived an appropriate value for 2 in terms of v the single path root mcan square phase rate r half the long time average mean square pressure at the receiver and po the threshold level using this equivalent value for t we observe that our exact and approximate detection models exhibit similar long term behavior but markedly different short term characteristics as compared with th t a model this is due to the memory of the process a property that cannot be accounted for in the t a model a comparison of these models with data obtained from various field experiments demonstrates in most cases an improved capability over the 2 a model

the total time on test plot and the cumulative total time on test statistic for a counting process results on the total time on test plot are usually obtained on the assumption that the number of events to be observed is fixed in advance here it is shown that the same large sample results hold when the number of events is random if a simple condition is satisfied

smooth goodness of fit tests for composite hypothesis in hazard based models consider a counting process n t t j with compensator process a t t j where a t 0 t y s 0 s ds y t t j is an observable predictable process and 0 is an unknown hazard rate function a general procedure for extending neyman s smooth goodness of fit test for the composite null hypothesis h 0 0 c 0 r q is proposed and developed the extension is obtained by embedding c in the class a k whose members are of the form 0 exp t x r k where 1 k t is a vector of observable random processes satisfying certain regularity conditions the tests are based on quadratic forms of the statistic 0 t s dm s where m t n t 0 t y s 0 s ds and is a restricted maximum likelihood estimator of asymptotic properties of the test statistics are obtained under a sequence of local alternatives and the asymptotic local powers of the tests are examined the effect of estimating by is ascertained and the problem of choosing the process is discussed the procedure is illustrated by developing tests for testing that 0 belongs to i the class of constant hazard rates and ii the class of weibull hazard rates with particular emphasis on the random censorship model simulation results concerning the achieved levels and powers of the tests are presented and the procedures are applied to three data sets that have been considered in the literature

shorter communication evaluating tests for increasing intensity of a poisson process the problem of testing the hypothesis of a constant intensity against the alternative of an increasing intensity function in a nonhomogeneous poisson process is studied and six wellknown tests are considered three of the six are shown to have desirable power function properties whereas the other three do not have the properties in fact the tests without the desirable properties should not be used nor should any nonparametric test be used

monotone smoothing with application to dose response curves and the assessment of synergism a spline based procedure for monotone curve smoothing is proposed and is illustrated by application to dose response curves it is then shown how such smoothing can be applied to assess possible synergism or antagonism of two drugs

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

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

closed form approximations and series representations for partially saturated ocean acoustic processes an approximate closed form expression for the value of the integral encountered in the calculation of the probability density function pdf of the envelope of a partially saturated ocean acoustic process is obtained furthermore an expression of this pdf as a series of modified bessel functions is presented the results may also be directly applied to the evaluation of the pdf encountered in the structural reliability analysis of rotating machinery components numerical applications show that the closed form expression is always within 1 2 of the exact result the required computational effort is substantially lower than that required by direct numerical integration

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

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

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

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

some simple robust methods for the analysis of recurrent events nelson discussed a method of estimating the cumulative mean function for identically distributed processes of recurrent events we show that a similar approach can be used with more general models including regression the key idea is to use point estimates based on poisson models and to develop robust variance estimates that are valid more generally the methods are illustrated on reliability and warranty data

second order statistics and discrete time detection modeling for partially saturated processes the basic problem in underwater detection is formulated under the general assumptions of partially saturated propagation of narrow band and acoustic signals expressions for the joint probability density function pdf of the short time average root mean square pressure at the receiver are obtained this joint pdf is a general result reducing to the pdf s for the fully saturated and the unsaturated cases for limiting values of the appropriate variables subsequently defining detection as occurring whenever exceeds a specified threshold level 0 and using the above results the upcrossing and downcrossing statistics of the envelope process are studied closed form expressions for the probability mass functions pmf s of the interarrival time time between two successive detections and holding time time between an upcrossing and the first subsequent downcrossing are obtained results using our partially saturated detection model reduce in limiting cases to results already obtained in the literature for fully saturated and unsaturated propagation

bayesian cart model search abstract in this article we put forward a bayesian approach for finding classification and regression tree cart models the two basic components of this approach consist of prior specification and stochastic search the basic idea is to have the prior induce a posterior distribution that will guide the stochastic search toward more promising cart models as the search proceeds such models can then be selected with a variety of criteria such as posterior probability marginal likelihood residual sum of squares or misclassification rates examples are used to illustrate the potential superiority of this approach over alternative methods

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

consistent detection of a monotonic trend superposed on a stationary time series summary consider a time series made up of a signal and a stationary autocorrelated error series a statistic is proposed for examining the hypothesis that the signal term is constant versus the hypothesis that it is monotonic in time the statistic is the ratio of a linear combination of the time series values with coefficients introduced by abelson tukey 1963 to an estimate of the standard error of the linear combination the statistic has asymptotic power 1 for a broad class of monotonic alternatives the procedure is illustrated for the series of river heights at a location on the rio negro in brazil where there is concern that the height is rising due to deforestation of the amazon basin the significance level obtained is 0025

monotonic calibrated interpolated look up tables real world machine learning applications may have requirements beyond accuracy such as fast evaluation times and interpretability in particular guaranteed monotonicity of the learned function with respect to some of the inputs can be critical for user confidence we propose meeting these goals for low dimensional machine learning problems by learning flexible monotonic functions using calibrated interpolated look up tables we extend the structural risk minimization framework of lattice regression to monotonic functions by adding linear inequality constraints in addition we propose jointly learning interpretable calibrations of each feature to normalize continuous features and handle categorical or missing data at the cost of making the objective non convex we address large scale learning through parallelization mini batching and random sampling of additive regularizer terms case studies on real world problems with up to sixteen features and up to hundreds of millions of training samples demonstrate the proposed monotonic functions can achieve state of the art accuracy in practice while providing greater transparency to users

some new tests of the power law process the power law process plp is often used to model failure data from repairable systems and its adequacy is tested with goodness of fit tests such as the cramer von mises test no single test however can be sensitive to all possible deviations from the plp exact tests that are powerful against some departures from the plp of particular interest to practitioners are derived heuristically the computation of their p values is described and their use is illustrated with examples one test is a modified version of the laplace test for trend another a modification of greenwood s test for clustering it is shown that these tests can be derived as score tests which ensures that they are powerful for large samples

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

simple parametric and nonparametric models for excess and relative mortality this paper studies two classes of hazard rate based models for the mortality in a group of individuals taking normal life expectancy into account in a multiplicative hazard model the estimate for the relative mortality generalises the standardised mortality ratio and the adequacy of a model with constant relative mortality can be tested using a type of total time on test statistic in an additive hazard model continuous time generalisations of a corrected survival curve and a normal survival curve are obtained and the adequacy of a model with constant excess mortality can again be tested using a type of total time on test statistic a model including both the multiplicative hazard model and the additive hazard model is briefly considered the use of the models is illustrated on a set of data concerning survival after operation for malignant melanoma

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

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

discrete time detection modeling for unsaturated ocean acoustic propagation the basic problem in ocean acoustic detection is formulated under the assumption of unsaturated sound propagation the latter essentially amounts to a constant signal plus gaussian noise detection is defined as occurring whenever the root mean square pressure at the receiver exceeds a specified threshold level 0 a two state discrete time markov model is derived and closed form expressions for the probability mass functions of the number of time steps separating two successive detections interarrival time or one detection and the first subsequent downcrossing holding time are presented expressions for the joint probability density function of at two different points in time are obtained and used to determine the relevant one step transition probabilities of the markov model sample results using the model are finally presented

testing for monotone trend in recurrent event processes tests for the presence of monotone trends in recurrent event processes have been proposed and studied by many authors however a general concept of trend is elusive and the dependence of the behavior of tests on the assumed definitions of no trend and trend and on the observation periods for the processes has been largely ignored we discuss these issues and study them through analytical results and simulation studies the article also presents robust tests for trend across multiple processes extends them to deal with interval censored event times and compares them and other well known trend tests with respect to the issues mentioned

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

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

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

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

additive isotonic models abstract additive isotonic models generalize linear models by replacing lines with isotonic nondecreasing transformations fitted transformations of several explanatory variables are added together and then transformed by a known function to yield fitted values of the response variable the isotonic transformations are chosen to minimize an explicit criterion such as the negative log likelihood by an algorithm that optimizes one transformation at a time while adjusting for the current fitted values of the others cycling until the criterion converges this approach can be used in various situations notably for generalizing ordinary linear regression and linear logistic regression at each step of the algorithm the needed optimal isotonic transformation is found using a simple generalization of the standard pool adjacent violators algorithm ayer brunk ewing reid and silverman 1955 the fitted transformations are always made up of flat steps so the technique is useful for finding optimal strati

evaluating tests for increasing intensity of a poisson process the problem of testing the hypothesis of a constant intensity against the alternative of an increasing intensity function in a nonhomogeneous poisson process is studied and six wellknown tests are considered three of the six are shown to have desirable power function properties whereas the other three do not have the properties in fact the tests without the desirable properties should not be used nor should any nonparametric test be used unbiasedness

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

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

repeated events and total time on test we adopt the total time on test procedure to investigate monotone time trends in the intensity in a repeated event setting the correct model is assumed to be a proportional hazards model with a random effect to account for dependence within subjects the method offers a simple routine for testing relevant hypotheses for recurrent event processes without making distributional assumptions about the frailty such assumptions may severely affect conclusions concerning regression coefficients and cause bias in the estimated heterogeneity the method is illustrated by re analyzing danish registry data and a long term swiss clinical study on recurrence in affective disorder

robust empirical bayes analyses of event rates a collection of i similar items generates point event histories for example machines experience failures or operators make mistakes suppose the intervals between events are modeled as iid exponential i or the counts as poisson i t i for the ith item furthermore so as to represent between item variability each individual rate parameter i is presumed drawn from a fixed super population with density g being a vector parameter a parametric empirical bayes peb setup for g specified alternatively as log student t n or gamma we exhibit the results of numerical procedures for estimating superpopulation parameters ll and for describing pooled estimates of the individual rates i obtained via bayes s formula three data sets are analyzed and convenient explicit approximate formulas are furnished for i estimates in the student t case the individual estimates are seen to have a robust quality

semiparametric regression for the mean and rate functions of recurrent events the counting process with the cox type intensity function has been commonly used to analyse recurrent event data this model essentially assumes that the underlying counting process is a time transformed poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process recently pepe and cai and lawless and co workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the poisson type assumption in this paper we provide a rigorous justification of such robust procedures through modern empirical process theory furthermore we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean rate model the advantages of the robust procedures are demonstrated through simulation studies an illustration with multiple infection data taken from a clinical study on chronic granulomatous disease is also provided

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