bayesian and non bayesian evidential updating abstract four main results are arrived at in this paper 1 closed convex sets of classical probability functions provide a representation of belief that includes the representations provided by shafer probability mass functions as a special case 2 the impact of uncertain evidence can be formally represented by dempster conditioning in shafer s framework 3 the impact of uncertain evidence can be formally represented in the framework of convex sets of classical probabilities by classical conditionalization 4 the probability intervals that result from dempster shafer updating on uncertain evidence are included in and may be properly included in the intervals that result from bayesian updating on uncertain evidence

total variation adaptive total variation and nonconvex smoothly clipped absolute deviation penalty for denoising blocky images the total variation based image denoising model has been generalized and extended in numerous ways improving its performance in different contexts we propose a new penalty function motivated by the recent progress in the statistical literature on high dimensional variable selection using a particular instantiation of the majorization minimization algorithm the optimization problem can be efficiently solved and the computational procedure realized is similar to the spatially adaptive total variation model our two pixel image model shows theoretically that the new penalty function solves the bias problem inherent in the total variation model the superior performance of the new penalty function is demonstrated through several experiments our investigation is limited to blocky images which have small total variation

inferences from multinomial data learning about a bag of marbles a new method is proposed for making inferences from multinomial data in cases where there is no prior information a paradigm is the problem of predicting the colour of the next marble to be drawn from a bag whose contents are initially completely unknown in such problems we may be unable to formulate a sample space because we do not know what outcomes are possible this suggests an invariance principle inferences based on observations should not depend on the sample space in which the observations and future events of interest are represented objective bayesian methods do not satisfy this principle this paper describes a statistical model called the imprecise dirichlet model for drawing coherent inferences from multinomial data inferences are expressed in terms of posterior upper and lower probabilities the probabilities are initially vacuous reflecting prior ignorance but they become more precise as the number of observations increases this model does satisfy the invariance principle two sets of data are analysed in detail in the first example one red marble is observed in six drawings from a bag inferences from the imprecise dirichlet model are compared with objective bayesian and frequentist inferences the second example is an analysis of data from medical trials which compared two treatments for cardiorespiratory failure in newborn babies there are two problems to draw conclusions about which treatment is more effective and to decide when the randomized trials should be terminated this example shows how the imprecise dirichlet model can be used to analyse data in the form of a contingency table

the system point method in exponential queues a level crossing approach the purpose of this paper is to describe the new system point method for analyzing queues it considers the stationary probability distribution of the waiting time in variations of the m m r queue with first come first served discipline for a large class of service mechanisms it is shown that the stationary probability density function of the waiting time evaluated at w 0 can be interpreted as the long run average of the number of times that the virtual wait becomes less than w per unit time theorems are presented which establish this interpretation of the probability density function of the virtual waiting time in terms of point processes generated by level crossings in the state space these theorems in combination with the principle of stationary set balance generate a system of model equations that can be written directly in addition the forms of these equations are often linear volterra integral equations of the second kind with parameter which yield direct analytical solutions an analogous theorem is proved for a variant of m g 1 two illustrative examples are presented

edge preserving and noise reducing reconstruction for magnetic particle imaging magnetic particle imaging mpi is an emerging medical imaging modality which is based on the non linear response of magnetic nanoparticles to an applied magnetic field it is an important feature of mpi that even fast dynamic processes can be captured for 3d volumes the high temporal resolution in turn leads to large amounts of data which have to be handled efficiently but as the system matrix of mpi is non sparse the image reconstruction gets computationally demanding therefore currently only basic image reconstruction methods such as tikhonov regularization are used however tikhonov regularization is known to oversmooth edges in the reconstructed image and to have only a limited noise reducing effect in this work we develop an efficient edge preserving and noise reducing reconstruction method for mpi as regularization model we propose to use the nonnegative fused lasso model and we devise a discretization that is adapted to the acquisition geometry of the preclinical mpi scanner considered in this work we develop a customized solver based on a generalized forward backward scheme which is particularly suitable for the dense and not well structured system matrices in mpi already a non optimized prototype implementation processes a 3d volume within a few seconds so that processing several frames per second seems amenable we demonstrate the improvement in reconstruction quality over the state of the art method in an experimental medical setup for an in vitro angioplasty of a stenosis

the variable selection problem abstract the problem of variable selection is one of the most pervasive model selection problems in statistical applications often referred to as the problem of subset selection it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors but there is uncertainty about which subset to use this vignette reviews some of the key developments that have led to the wide variety of approaches for this problem

the ii p method for estimating multivariate functions from noisy data the method for estimating an underlying smooth function of m variables x l xm using noisy data is based on approximating it by a sum of products of the form m m x m the problem is then reduced to estimating the univariate functions in the products a convergent algorithm is described the method keeps tight control on the degrees of freedom used in the fit many examples are given the quality of fit given by the method is excellent usually only a few products are enough to fit even fairly complicated functions the coding into products of univariate functions allows a relatively understandable interpretation of the multivariate fit

local extremes runs strings and multiresolution the paper considers the problem of nonparametric regression with emphasis on controlling the number of local extremes two methods the run method and the taut string multiresolution method are introduced and analyzed on standard test beds it is shown that the number and locations of local extreme values are consistently estimated rates of convergence are proved for both methods the run method converges slowly but can withstand blocks as well as a high proportion of isolated outliers the rate of convergence of the taut string multiresolution method is almost optimal the method is extremely sensitive and can detect very low power peaks section 1 contains an introduction with special reference to the number of local extreme values the run method is described in section 2 and the taut string multiresolution method in section 3 low power peaks are considered in section 4 section 5 contains a comparison with other methods and section 6 a short conclusion the proofs are given in section 7 and the taut string algorithm is described in the appendix

on the fly approximation of multivariate total variation minimization in the context of change point detection addressed by total variation minimization strategies an efficient on the fly algorithm has been designed leading to exact solutions for univariate data in this contribution an extension of such an on the fly strategy to multivariate data is investigated the proposed algorithm relies on the local validation of the karush kuhn tucker conditions on the dual problem showing that the non local nature of the multivariate setting precludes to obtain an exact on the fly solution we devise an on the fly algorithm delivering an approximate solution whose quality is controlled by a practitioner tunable parameter acting as a trade off between quality and computational cost performance assessment shows that high quality solutions are obtained on the fly while benefiting of computational costs several orders of magnitude lower than standard iterative procedures the proposed algorithm thus provides practitioners with an efficient multivariate change point detection on the fly procedure

wild binary segmentation for multiple change point detection we propose a new technique called wild binary segmentation wbs for consistent estimation of the number and locations of multiple change points in data we assume that the number of change points can increase to infinity with the sample size due to a certain random localisation mechanism wbs works even for very short spacings between the change points and or very small jump magnitudes unlike standard binary segmentation on the other hand despite its use of localisation wbs does not require the choice of a window or span parameter and does not lead to a significant increase in computational complexity wbs is also easy to code we propose two stopping criteria for wbs one based on thresholding and the other based on what we term the strengthened schwarz information criterion we provide default recommended values of the parameters of the procedure and show that it offers very good practical performance in comparison with the state of the art the wbs methodology is implemented in the r package wbs available on cran in addition we provide a new proof of consistency of binary segmentation with improved rates of convergence as well as a corresponding result for wbs

parameter estimation for differential equations a generalized smoothing approach summary we propose a new method for estimating parameters in models that are defined by a system of non linear differential equations such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation the paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non linear differential equations the approach is based on a modification of data smoothing methods along with a generalization of profiled estimation we derive estimates and confidence intervals and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology the performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

tail greedy bottom up data decompositions and fast multiple change point detection this article proposes a tail greedy bottom up transform for one dimensional data which results in a nonlinear but conditionally orthonormal multiscale decomposition of the data with respect to an adaptively chosen unbalanced haar wavelet basis the tail greediness of the decomposition algorithm whereby multiple greedy steps are taken in a single pass through the data both enables fast computation and makes the algorithm applicable in the problem of consistent estimation of the number and locations of multiple changepoints in data the resulting agglomerative change point detection method avoids the disadvantages of the classical divisive binary segmentation and offers very good practical performance it is implemented in the r package breakfast available from cran

adaptive nonparametric regression with the k nearest neighbour fused lasso the fused lasso also known as total variation denoising is a locally adaptive function estimator over a regular grid of design points in this article we extend the fused lasso to settings in which the points do not occur on a regular grid leading to a method for nonparametric regression this approach which we call the formula see text nearest neighbours fused lasso involves computing the formula see text nearest neighbours graph of the design points and then performing the fused lasso over this graph we show that this procedure has a number of theoretical advantages over competing methods specifically it inherits local adaptivity from its connection to the fused lasso and it inherits manifold adaptivity from its connection to the formula see text nearest neighbours approach in a simulation study and an application to flu data we show that excellent results are obtained for completeness we also study an estimator that makes use of an formula see text graph rather than a formula see text nearest neighbours graph and contrast it with the formula see text nearest neighbours fused lasso

bayesian selection for the ell _2 potts model regularization parameter 1 d piecewise constant signal denoising piecewise constant denoising can be solved either by deterministic optimization approaches based on the potts model or by stochastic bayesian procedures the former lead to low computational time but require the selection of a regularization parameter whose value significantly impacts the achieved solution and whose automated selection remains an involved and challenging problem conversely fully bayesian formalisms encapsulate the regularization parameter selection into hierarchical models at the price of high computational costs this contribution proposes an operational strategy that combines hierarchical bayesian and potts model formulations with the double aim of automatically tuning the regularization parameter and maintaining computational efficiency the proposed procedure relies on formally connecting a bayesian framework to a ell 2 potts functional behaviors and performance for the proposed piecewise constant denoising and regularization parameter tuning techniques are studied qualitatively and assessed quantitatively and shown to compare favorably against those of a fully bayesian hierarchical procedure both in accuracy and computational load

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

total bounded variation regularization as a bilaterally constrained optimization problem it is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem existence of a lagrange multiplier and an optimality system are established this allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations here in particular the primal dual active set method considered as a semismooth newton method is analyzed and superlinear convergence is proved as a by product we obtain that the lagrange multiplier associated with the box constraints acts as an edge detector numerical results for image denoising and zooming resizing show the efficiency of the new approach

the limitation of bayesianism in the current discussion about the capacity of bayesianism in reasoning under uncertainty there is a conceptual and notational confusion between the explicit condition and the implicit condition of a probability evaluation consequently the limitation of bayesianism is often seriously underestimated to represent the uncertainty of a belief system where revision is needed it is not enough to assign a probability value to each belief

optimal and simple approximate solutions to a production inventory system with stochastic and deterministic demand abstract we consider a continuous review production inventory system where demand is a mixture of a deterministic component and a random component which follows a compound poisson process demand is satisfied by a production facility which may be in production or idle while in production the facility produces at a constant rate a two number s s policy is used to control production when inventory reaches level s production is turned off and when inventory drops below level s production is turned on a level crossing approach is used to derive the steady state distribution of inventory level from which the exact total expected cost function consisting of setup inventory holding and backorder costs is determined the optimal s s policy can be found through a search of the expected cost function we propose approximations that give simple closed form solutions with near optimal performance

mode identification of volatility in time varying autoregression in many applications time series exhibit nonstationary behavior that might reasonably be modeled as a time varying autoregressive ar process in the context of such a model we discuss the problem of testing for modality of the variance function we propose a test of modality that is local and when used iteratively can be used to identify the total number of modes in a given series this problem is closely related to peak detection and identification which has applications in many fields we propose a test that under appropriate assumptions is asymptotically distribution free under the null hypothesis even though nonparametric estimation of the ar parameter functions is involved simulation studies and applications to real datasets illustrate the behavior of the test

consistencies and rates of convergence of jump penalized least squares estimators we study the asymptotics for jump penalized least squares regression aiming at approximating a regression function by piecewise constant functions besides conventional consistency and convergence rates of the estimates in l 2 0 1 our results cover other metrics like skorokhod metric on the space of cadlag functions and uniform metrics on c 0 1 we will show that these estimators are in an adaptive sense rate optimal over certain classes of approximation spaces special cases are the class of functions of bounded variation piecewise holder continuous functions of order 0 1 and the class of step functions with a finite but arbitrary number of jumps in the latter setting we will also deduce the rates known from change point analysis for detecting the jumps finally the issue of fully automatic selection of the smoothing parameter is addressed

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

production inventory systems with lost sales and compound poisson demands this paper considers a continuous review single product production inventory system with a constant replenishment rate compound poisson demands and lost sales two objective functions that represent metrics of operational costs are considered 1 the sum of the expected discounted inventory holding costs and lost sales penalties both over an infinite time horizon given an initial inventory level and 2 the long run time average of the same costs the goal is to minimize these cost metrics with respect to the replenishment rate it is however not possible to obtain closed form expressions for the aforementioned cost functions directly in terms of positive replenishment rate prr to overcome this difficulty we construct a bijection from the prr space to the space of positive roots of lundberg s fundamental equation to be referred to as the lundberg positive rootlpr space this transformation allows us to derive closed form expressions for the aforementioned cost metrics with respect to the lpr variable in lieu of the prr variable we then proceed to solve the optimization problem in the lpr space and finally recover the optimal replenishment rate from the optimal lpr variable via the inverse bijection for the special cases of constant or loss proportional penalty and exponentially distributed demand sizes we obtain simpler explicit formulas for the optimal replenishment rate

a production inventory model with randomly changing environmental conditions we consider a limited capacity production inventory system with linear production rate and compound poisson demand in which both production and demand processes are subject to independently and randomly changing environmental conditions assuming that these conditions are represented by two continuous time homogeneous markov chains and shortages are lost we derive the limiting distribution of the inventory level and present some numerical results in terms of the ensuing performance measures

the compensation method applied to a one product production inventory problem this paper considers a one product one machine production inventory problem demand requests for the product are governed by a poisson process with demand size being an exponential random variable the production facility may be in production or idle while in production the facility produces continuously at a constant rate the objective is to minimize system costs consisting of setup costs inventory holding costs and backorder costs given a two critical number policy the problem is analyzed as a constrained markov process using the compensation method a closed form expression is found for system cost as a function of the policy parameters this cost function may then be searched to find an optimal policy the production inventory problem is related to a server backlog process for an m m 1 system in which the server departs when the system empties and returns when the backlog exceeds a specified threshold the application of the compensation method to this server backlog process is discussed

densities spectral densities and modality this paper considers the problem of specifying a simple approximating density function for a given data set x 1 x n simplicity is measured by the number of modes but several different definitions of approximation are introduced the taut string method is used to control the numbers of modes and to produce candidate approximating densities refinements are introduced that improve the local adaptivity of the procedures and the method is extended to spectral densities

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

using simulated annealing to optimize the feature selection problem in marketing applications the feature selection also specification problem is concerned with finding the most influential subset of predictors in predictive modeling from a much larger set of potential predictors that can contain hundreds of predictors the problem belongs to the realm of combinatorial optimization where the objective is to find the subset of variables that optimize the value of some goodness of fit function due to the dimensionality of the problem the feature selection problem belongs to the group of np hard problems most of the available predictors are noisy or redundant and add very little if any to the prediction power of the model using all the predictors in the model often results in strong over fitting and very poor predictions constructing a prediction model by checking out all possible subsets is impractical due to computational volume looking on the contribution of each predictor separately is not accurate because it ignores the inter correlations between predictors as a result no analytic solution is available for the feature selection problem requiring that one resorts to heuristics in this paper we employ the simulated annealing sa approach which is one of the leading stochastic search methods for specifying a large scale linear regression model the sa results are compared to the results of the more common stepwise regression swr approach for model specification the models are applied on realistic data sets in database marketing we also use simulated data sets to investigate what data characteristics make the swr approach equivalent to the supposedly more superior sa approach

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

unbalanced haar technique for nonparametric function estimation the discrete unbalanced haar uh transform is a decomposition of one dimensional data with respect to an orthonormal haar like basis where jumps in the basis vectors do not necessarily occur in the middle of their support we introduce a multiscale procedure for estimation in gaussian noise that consists of three steps a uh transform thresholding of the decomposition coefficients and the inverse uh transform we show that our estimator is mean squared consistent with near optimal rates for a wide range of functions uniformly over uh bases that are not too unbalanced a vital ingredient of our approach is basis selection we choose each basis vector so that it best matches the data at a specific scale and location where the latter parameters are determined by the parent basis vector our estimator is computable in o n log n operations a simulation study demonstrates the good performance of our estimator compared with state of the art competitors we apply our method to the estimation of the mea

properties and refinements of the fused lasso we consider estimating an unknown signal both blocky and sparse which is corrupted by additive noise we study three interrelated least squares procedures and their asymptotic properties the first procedure is the fused lasso put forward by friedman et al ann appl statist 1 2007 302 332 which we modify into a different estimator called the fused adaptive lasso with better properties the other two estimators we discuss solve least squares problems on sieves one constrains the maximal l 1 norm and the maximal total variation seminorm and the other restricts the number of blocks and the number of nonzero coordinates of the signal we derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso and we derive convergence rates for the sieve estimators explicitly in terms of the constraining parameters

multiscale change point inference summary we introduce a new estimator the simultaneous multiscale change point estimator smuce for the change point problem in exponential family regression an unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level the probability of overestimating the true number of change points k is controlled by the asymptotic null distribution of the multiscale test statistic further we derive exponential bounds for the probability of underestimating k by balancing these quantities will be chosen such that the probability of correctly estimating k is maximized all results are even non asymptotic for the normal case on the basis of these bounds we construct asymptotically honest confidence sets for the unknown step function and its change points at the same time we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log term finally the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n even for an unbounded number of change points we illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions the performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy

splines in higher order tv regularization splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces in this paper we show in a strictly discrete setting that splines of degree m 1 solve also a minimization problem with quadratic data term and m th order total variation tv regularization term in contrast to problems with quadratic regularization terms involving m th order derivatives the spline knots are not known in advance but depend on the input data and the regularization parameter more precisely the spline knots are determined by the contact points of the m th discrete antiderivative of the solution with the tube of width 2 around the m th discrete antiderivative of the input data we point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the sobolev space w 2 0 m from this point of view the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

sporadic and continuous clearing policies for a production inventory system under an m g demand process a production inventory system is filled continuously at rate 1 and satisfies demands of i i d random sizes that arrive at poisson times for this system we consider two clearing policies under sporadic review clearing takes place after a random time independent of the content process under continuous review the system is cleared as soon as the inventory level reaches some pre specified threshold we derive explicit results for the associated expected discounted cost functionals under both types of clearing

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

optimal spatial adaptation to inhomogeneous smoothness an approach based on kernel estimates with variable bandwidth selectors a new v ria e 1 s e 0r estimation is proposed the application of this bandwidth selector leads to kernel estimates that achieve optimal rates of convergence over b 3 sses this implies that the procedure adapts to spatially inhomogeneous s noothness in particu lar the estimates share optimality properties with wavelet estimates based on thresholding of empirical wavelet coefficients 1 introduction in nonparametric curve estimation the statistical anal ysis may focus on the inference of the qualitative structure of the analyzed curve often interesting features of the curve are connected with spatially inhomogeneous smoothness in this case curve estimates that are spatially adaptlve ar e appropriaie a variety of such procedures have been proposed in the literature in breiman friedman olshen and stone 1983 piecewise constant least squares estimates are considered with a data adaptive choice of the pieces cart more generally friedman and silverman 1989 friedman 1991 and luo and wahba 1995 use variable knot splines mars knot points are added removed and allocated recursively using cross validation techniques these methods have shown good performance in simulations and real data examples however no asymptotic theory is available mammen and van de geer 1997 discuss penalized least squares curve estimation for spatially inhomogeneous curves they propose penalty terms which allow more spatial inhomogeneity than the usual l2 norms of deriva tives of the curve the estimates turn out to be variable knot splines see also mammen 1991 results on rates of convergence and a pointwise asymptotic distribution theory are given miiller and stadtmiiller 1987 staniswalis 1989 and brockmann gasser and hermann 1993 propose kernel estimation with locally variable band width selectors the calculation of local bandwidths is based on pilot estima tion of local smoothness characteristics an asymptotic analysis is available

the dfs fused lasso linear time denoising over general graphs the fused lasso also known as anisotropic total variation denoising is widely used for piecewise constant signal estimation with respect to a given undirected graph the fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure but for a special graph structure namely the chain graph the fused lasso or simply 1d fused lasso can be computed in linear time in this paper we revisit a result recently established in the online classification literature herbster et al 2009 cesa bianchi et al 2013 and show that it has important implications for signal denoising on graphs the result can be translated to our setting as follows given a general graph if we run the standard depth first search dfs traversal algorithm then the total variation of any signal over the chain graph induced by dfs is no more than twice its total variation over the original graph this result leads to several interesting theoretical and computational conclusions letting m and n denote the number of edges and nodes respectively of the graph in consideration it implies that for an underlying signal with total variation t over the graph the fused lasso properly tuned achieves a mean squared error rate of t2 3n 2 3 moreover precisely the same mean squared error rate is achieved by running the 1d fused lasso on the dfs induced chain graph importantly the latter estimator is simple and computationally cheap requiring o m operations to construct the dfs induced chain and o n operations to compute the 1d fused lasso solution over this chain further for trees that have bounded maximum degree the error rate of t2 3n 2 3 cannot be improved in the sense that it is the minimax rate for signals that have total variation t over the tree finally several related results also hold for example the analogous result holds for a roughness measure defined by the l0 norm of differences across edges in place of the total variation metric

estimating a density under order restrictions nonasymptotic minimax risk on considere la classe de toutes les densites unimodales definies sur un intervalle de longueur l et borne par h on etudie le risque minimax sur cette classe lorsqu on estime en utilisant n observations i i d la perte etant mesuree par la distance l 1 entre l estimateur et la densite reelle

quantile smoothing splines summary although nonparametric regression has traditionally focused on the estimation of conditional mean functions nonparametric estimation of conditional quantile functions is often of substantial practical interest we explore a class of quantile smoothing splines defined as solutions to