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

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

splines as local smoothers a smoothing spline is a nonparametric curve estimate that is defined as the solution to a minimization problem one problem with this representation is that it obscures the fact that a spline like most other nonparametric estimates is a local weighted average of the observed data this property has been used extensively to study the limiting properties of kernel estimates and it is advantageous to apply similar techniques to spline estimates although equivalent kernels have been identified for a smoothing spline these functions are either not accurate enough for asymptotic approximations or are restricted to equally spaced points this paper extends this previous work to understand a spline estimate s local properties it is shown that the absolute value of the spline weight function decreases exponentially away from its center this result is not asymptotic the only requirement is that the empirical distribution of the observation points be sufficiently close to a continuous distribution with a strictly positive density function these bounds are used to derive the asymptotic form for the bias and variance of a first order smoothing spline estimate the arguments leading to this result can be easily extended to higher order splines

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

wavelet thresholding for non necessarily gaussian noise idealism for various types of noise exponential normal mixture compactly supported wavelet thresholding methods are studied problems linked to the existence of optimal thresholds are tackled and minimaxity properties of the methods also analyzed a coefficient dependent method for choosing thresholds is also briefly presented

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

the role of pseudo data for robust smoothing with application to wavelet regression summary we propose a robust curve and surface estimator based on m type estimators and penalty based smoothing this approach also includes an application to wavelet regression the concept of pseudo data a transformation of the robust additive model to the one with bounded errors is used to derive some theoretical properties and also motivate a computational algorithm the resulting algorithm termed the es algorithm is computationally fast and provides a simple way of choosing the amount of smoothing moreover it is easily described straightforwardly implemented and can be extended to other wavelet regression settings such as irregularly spaced data and image denoising results from a simulation study and real data examples demonstrate the promising empirical properties of the proposed approach

stepup procedures controlling generalized fwer and generalized fdr in many applications of multiple hypothesis testing where more than one false rejection can be tolerated procedures controlling error rates measuring at least k false rejections instead of at least one for some fixed k 1 can potentially increase the ability of a procedure to detect false null hypotheses the k fwer a generalized version of the usual familywise error rate fwer is such an error rate that has recently been introduced in the literature and procedures controlling it have been proposed a further generalization of a result on the k fwer is provided in this article in addition an alternative and less conservative notion of error rate the k fdr is introduced in the same spirit as the k fwer by generalizing the usual false discovery rate fdr a k fwer procedure is constructed given any set of increasing constants by utilizing the kth order joint null distributions of the p values without assuming any specific form of dependence among all the p values procedures controlling the k fdr are also developed by using the kth order joint null distributions of the p values first assuming that the sets of null and nonnull p values are mutually independent or they are jointly positively dependent in the sense of being multivariate totally positive of order two mtp2 and then discarding that assumption about the overall dependence among the p values

false discovery control for random fields this article extends false discovery rates to random fields for which there are uncountably many hypothesis tests we develop a method for finding regions in the field s domain where there is a significant signal while controlling either the proportion of area or the proportion of clusters in which false rejections occur the method produces confidence envelopes for the proportion of false discoveries as a function of the rejection threshold from the confidence envelopes we derive threshold procedures to control either the mean or the specified tail probabilities of the false discovery proportion an essential ingredient of this construnction is a new algorithm to compute a confidence superset for the set of all true null locations we demonstrate our method with applications to scan statistics and functional neuroimaging

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

level adjusted funnel plots based on predicted marginal expectations an application to prophylactic antibiotics in gallstone surgery funnel plots are widely used to visualize grouped data for example in institutional comparison this paper extends the concept to a multi level setting displaying one level at a time adjusted for the other levels as well as for covariates at all levels these level adjusted funnel plots are based on a markov chain monte carlo fit of a random effects model translating the estimated model parameters to predicted marginal expectations working within the estimation framework we accommodate outlying institutions using heavy tailed random effects distributions we also develop computer efficient methods to compute predicted probabilities in the case of dichotomous outcome data and various random effect distributions we apply the method to a data set on prophylactic antibiotics in gallstone surgery

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

degenerate routes to chaos we present examples of simple dynamical systems containing alternative routes to chaos which are degenerate with a period doubling route the existence of degenerate routes implies the possibility of having more entangled dynamical behaviors hidden in observations based on a single variable time series possibly requiring therefore special care to discriminate them a model system allowing the investigation of transitions between degenerate routes is also considered

adaptive m estimation in nonparametric regression a method for robust nonparametric regression is discussed a method for robust nonparametric regression is discussed we consider kernel m estimates of the regression function using huber s function and extend results of hardle and gasser to the case of random designs a practical adaptive procedure is proposed consisting of simultaneously minimising a cross validatory criterion with respect to both the smoothing parameter and a robustness parameter occurring in the function this method is shown to possess a theoretical asymptotic optimality property while some simulated examples confirm that the approach is practicable

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

a method for studying stability domains in physical models abstract we present a method for investigating the simultaneous movement of all zeros of equations of motions defined by discrete mappings the method is used to show that knowledge of the interplay of all zeros is of fundamental importance for establishing periodicities and relative stability properties of the various possible physical solutions the method is also used i to show that the frontiere set of fatou is defined primarily by zeros of functions leading to an entire invariant limiting function which underlies every dynamical system ii to identify cyclotomic polynomials as components of the limiting function obtained for a parameter value supporting a particular superstable orbit of the quadratic map iii to describe highly symmetric periodic cycles embedded in these components and iv to provide an unified picture about which mathematical objects form basin boundaries of dynamical systems in general the closure of all zeros not belonging to stable orbits

funnel plots for comparing institutional performance funnel plots are recommended as a graphical aid for institutional comparisons in which an estimate of an underlying quantity is plotted against an interpretable measure of its precision control limits form a funnel around the target outcome in a close analogy to standard shewhart control charts examples are given for comparing proportions and changes in rates assessing association between outcome and volume of cases and dealing with over dispersion due to unmeasured risk factors we conclude that funnel plots are flexible attractively simple and avoid spurious ranking of institutions into league tables

a direct approach to false discovery rates summary multiple hypothesis testing involves guarding against much more complicated errors than single hypothesis testing whereas we typically control the type i error rate for a single hypothesis test a compound error rate is controlled for multiple hypothesis tests for example controlling the false discovery rate fdr traditionally involves intricate sequential p value rejection methods based on the observed data whereas a sequential p value method fixes the error rate and estimates its corresponding rejection region we propose the opposite approach we fix the rejection region and then estimate its corresponding error rate this new approach offers increased applicability accuracy and power we apply the methodology to both the positive false discovery rate pfdr and fdr and provide evidence for its benefits it is shown that pfdr is probably the quantity of interest over fdr also discussed is the calculation of the q value the pfdr analogue of the p value which eliminates the need to set the error rate beforehand as is traditionally done some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the benjamini hochberg fdr method

robust wavelet denoising for extracting a signal from noisy data waveshrink and basis pursuit are powerful tools both from an empirical and asymptotic point of view they are especially efficient at estimating spatially inhomogeneous signals when the noise is gaussian their performance is altered when the noise has a long tail distribution for instance when outliers are present we propose a robust wavelet based estimator using a robust loss function this entails solving a nontrivial optimization problem and appropriately choosing the smoothing and robustness parameters we illustrate the advantage of the robust wavelet denoising procedure on simulated and real data

extending the scope of wavelet regression methods by coefficient dependent thresholding abstract various aspects of the wavelet approach to nonparametric regression are considered with the overall aim of extending the scope of wavelet techniques to irregularly spaced data to regularly spaced datasets of arbitrary size to heteroscedastic and correlated data and to data that contain outliers the core of the methodology is an algorithm for finding ail of the variances and within level covariances in the wavelet table of a sequence with given covariance structure if the original covariance matrix is band limited then the algorithm is linear in the length of the sequence the variance calculation algorithm allows data on any set of independent variable values to be treated by first interpolating to a fine regular grid of suitable length and then constructing a wavelet expansion of the gridded data various thresholding methods are discussed and investigated exact risk formulas for the mean square error of the methodology for given design are derived good performance is obtained by nois

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

multi state coupled map lattices abstract we investigate a two dimensional locally coupled map lattice cml with the local dynamics driven by the multi attractor quartic map in particular we explore a region where two local fixed points exist one being periodic and the other chaotic different sets of initial conditions such as random initial values for each site or arrangements favoring equal weights to the different local attractors were used the system reaches different asymptotic states as the intensity or the topology of the local coupling is varied among the asymptotic states we find either homogeneous collective behavior or mixtures of these with synchronized states these states are characterized and interpreted throughout this work by the distributions of the values of the maps and by the average roughness over the lattice

a comparison of a spline estimate to its equivalent kernel estimate it has been observed that to a smoothing spline operator there corresponds an equivalent kernel operator these two operators have been compared in a variety of norms cox 1984 silverman 1984 in this paper we refine the existing bounds for the particular case of the spline estimator considered in rice and rosenblatt 1983 and its corresponding equivalent kernel estimator we obtain detailed asymptotic expressions for the bias and covariance functions of the two estimates and provide rates of convergence direct comparison then shows that the two estimates are similar they may differ somewhat in their higher order boundary behavior