high dimensional classification via nonparametric empirical bayes and maximum likelihood inference we propose new nonparametric empirical bayes methods for high dimensional classification our classifiers are designed to approximate the bayes classifier in a hypothesized hierarchical model where the prior distributions for the model parameters are estimated nonparametrically from the training data as is common with nonparametric empirical bayes the proposed classifiers are effective in high dimensional settings even when the underlying model parameters are in fact nonrandom we use nonparametric maximum likelihood estimates of the prior distributions following the elegant approach studied by kiefer wolfowitz in the 1950s however our implementation is based on a recent convex optimization framework for approximating these estimates that is well suited for large scale problems we derive new theoretical results on the accuracy of the approximate estimator which help control the misclassification rate of one of our classifiers we show that our methods outperform several existing methods in simulations and perform well when gene expression microarray data is used to classify cancer patients

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 for monotonicity of a regression mean by calibrating for linear functions a new approach to testing for monotonicity of a regression mean not requiring computation of a curve estimator or a bandwidth is suggested it is based on the notion of running gradients over short intervals although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip excess mass approach for testing modality hypotheses about densities like the latter methods the new technique does not suffer difficulties caused by almost flat parts of the target function in fact it is calibrated so as to work well for flat response curves and as a result it has relatively good power properties in boundary cases where the curve exhibits shoulders in this respect as well as in its construction the running gradients approach differs from alternative techniques based on the notion of a critical bandwidth

a confidence limit for the empirical mode decomposition and hilbert spectral analysis the confidence limit is a standard measure of the accuracy of the result in any statistical analysis most of the confidence limits are derived as follows the data are first divided into subsections and then under the ergodic assumption the temporal mean is substituted for the ensemble mean next the confidence limit is defined as a range of standard deviations from this mean however such a confidence limit is valid only for linear and stationary processes furthermore in order for the ergodic assumption to be valid the subsections have to be statistically independent for non stationary and nonlinear processes such an analysis is no longer valid the confidence limit of the method here termed emd hsa for empirical mode decomposition hilbert spectral analysis is introduced by using various adjustable stopping criteria in the sifting processes of the emd step to generate a sample set of intrinsic mode functions imfs the emd technique acts as a pre processor for hsa on the original data producing a set of components imfs from the original data that equal the original data when added back together each imf represents a scale in the data from smallest to largest the ensemble mean and standard deviation of the imf sample sets obtained with different stopping criteria are calculated and these form a simple random sample set the confidence limit for emd hsa is then defined as a range of standard deviations from the ensemble mean without evoking the ergodic assumption subdivision of the data stream into short sections is unnecessary hence the results and the confidence limit retain the full frequency resolution of the full dataset this new confidence limit can be applied to the analysis of nonlinear and non stationary processes by these new techniques data from length of day measurements and a particularly violent recent earthquake are used to demonstrate how the confidence limit is obtained and applied by providing a confidence limit for this new approach a stable range of stopping criteria for the decomposition or sifting phase emd has been established making the results of the final processing with hsa and the entire emd hsa method more definitive

group linear empirical bayes estimates for a heteroscedastic normal mean abstractthe problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional empirical bayes estimators by taking a different approach that treats the known variances as part of the random observations we restore symmetry and thus the effectiveness of such methods we suggest a group linear empirical bayes estimator which collects observations with similar variances and applies a spherically symmetric estimator to each group separately the proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule and thus provides a more ambitious benchmark than that considered in the previous literature our estimator asymptotically achieves the new oracle risk under appropriate conditions and at the same time is minimax the group linear estimator is particularly advantageous in situations where the true means and observed variances are empirically dependent to demonstrate the meri

mass recentred kernel smoothers the local linear smoother usually gives better performance than the nadaraya watson smoother an exception is the case of data sparsity here we discuss a modification of the nadaraya watson smoother by muller song 1993 based on a horizontal shift of the kernel weights towards the local centre of mass of the design points this gives performance similar to the local linear when that works well and better performance when it does not the new smoother also preserves monotonicity shifting towards the centre of mass is also used to develop a modified kernel density estimate which cancels the well known peak spreading effect

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

applications of high resolution time frequency transforms to attenuation estimation abstractattenuation estimates quantify the loss of energy of propagating seismic waves due to anelastic processes it is often carried out in the frequency domain the most well known methods for attenuation estimation such as the spectral ratio and frequency shift methods compare spectral shapes of waveforms along a given raypath they require broad spectra such as those obtained with the fourier transform and the continuous wavelet transform these methods are incompatible with high resolution time frequency transforms which drastically localize time frequency information on the other hand these transforms indicate stronger resistance to noise and can be used in combination with the peak frequency method to estimate attenuation we have applied high resolution transforms namely the synchrosqueezing transform basis pursuit and complete ensemble empirical mode decomposition to a synthetic wedge example and two seismic data set examples a seismic reflection profile and a vertical seismic profile

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

on hilbert s inequality and its applications abstract in this paper the weight coefficient of the form n 2n 1 with n 0 is introduced improvements on hilbert s inequality and the hardy littlewood inequality are established and these results are extended

bayesian inference on quasi sparse count data there is growing interest in analysing high dimensional count data which often exhibit quasi sparsity corresponding to an overabundance of zeros and small nonzero counts existing methods for analysing multivariate count data via poisson or negative binomial log linear hierarchical models with zero inflation cannot flexibly adapt to quasi sparse settings we develop a new class of continuous local global shrinkage priors tailored to quasi sparse counts theoretical properties are assessed including flexible posterior concentration and stronger control of false discoveries in multiple testing simulation studies demonstrate excellent small sample properties relative to competing methods we use the method to detect rare mutational hotspots in exome sequencing data and to identify north american cities most impacted by terrorism

on the norm of a hilbert s type linear operator and applications in this paper the norm of a hilbert s type linear operator t l r l r r 1 r p q is given as applications a new operator inequality and the equivalent forms with the norm are obtained and particularly some new extended hilbert s type inequalities and the equivalent forms with the best constant factors are established

a class of grouped brunk estimators and penalized spline estimators for monotone regression we study a class of monotone univariate regression estimators we use b splines to approximate an underlying regression function and estimate spline coefficients based on grouped data we investigate asymptotic properties of two monotone estimators a grouped brunk estimator and a penalized monotone estimator these estimators are consistent at the boundary and their mean square errors achieve optimal convergence rates under suitable assumptions of the true regression function asymptotic distributions are developed and are shown to be independent of spline degrees and the number of knots simulation results and car data illustrate performance of the proposed estimators

distribution free tests of stochastic monotonicity this article proposes an omnibus test for monotonicity of nonpara metric conditional distributions and its moments unlike previous proposals our method does not require smooth estimation of the derivatives of nonparametric curves and it can be implemented even when probability densities do not exist in fact we only require continuity of the marginal distributions under the null and xed alternatives distinguishing features of our approach are that critical values are pivotal under the null in nite samples and the test is invariant to any monotonic continuous transformation of the explanatory variable the test sta tistic is the sup norm of the dierence between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate the resulting test is able to detect local alternatives converging to the null at the parametric rate n 1 2 with n the sample size the article also discusses several applications and extensions of the proposal these include testing monotonicity of general conditional moments and the extension to multivariate explanatory variables the nite sample performance of the test is examined by means of a monte carlo experiment

microseismic and seismic denoising via ensemble empirical mode decomposition and adaptive thresholding random and coherent noise exists in microseismic and seismic data and suppressing noise is a crucial step in seismic processing we have developed a novel seismic denoising method based on ensemble empirical mode decomposition eemd combined with adaptive thresholding a signal was decomposed into individual components called intrinsic mode functions imfs each decomposed signal was then compared with those imfs resulting from a white noise realization to determine if the original signal contained structural features or white noise only a thresholding scheme then removed all nonstructured portions our scheme is very flexible and it is applicable in a variety of domains or in a diverse set of data for instance it can serve as an alternative for random noise removal by band pass filtering in the time domain or spatial prediction filtering in the frequency offset domain to enhance the lateral coherence of seismic sections we have determined its potential for microseismic and reflection seismic denoising by comparing its performance on synthetic and field data using a variety of methods including band pass filtering basis pursuit denoising frequency offset deconvolution and frequency offset empirical mode decomposition

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

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

semi nonparametric estimation of the call option price surface under strike and time to expiry no arbitrage constraints abstract we suggest a semi nonparametric estimator for the call option price surface the estimator is a bivariate tensor product b spline to enforce no arbitrage constraints across strikes and expiry dates we establish sufficient no arbitrage conditions on the control net of the b spline surface the conditions are linear and therefore allow for an implementation of the estimator by means of standard quadratic programming techniques the consistency of the estimator is proved by means of simulations we explore the statistical efficiency benefits that are associated with estimating option price surfaces and state price densities under the full set of no arbitrage constraints we estimate a call option price surface families of first order strike derivatives and state price densities for s p 500 option data

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

on independent statistical decision problems and products of diffusions letzt be a null recurrent diffusion on p with generatorg 1 r r for smooth positiver this note constructs an independent recurrent diffusionz ton 1 such that zt z t is transient in p 1 this resolves negatively an old question in simultaneous estimation is there an admissible but not bayes estimator x of the mean of a multivariate normal distribution for quadratic loss with the property for every admissible x wherex is normal and independent ofx remains admissible in the combined problem obtained by summing the component losses

meta analysis with fixed unknown study specific parameters meta analysis is a valuable tool for combining information from independent studies however most common meta analysis techniques rely on distributional assumptions that are difficult if not impossible to verify for instance in the commonly used fixed effects and random effects models we take for granted that the underlying study level parameters are either exactly the same across individual studies or that they are realizations of a random sample from a population often under a parametric distributional assumption in this article we present a new framework for summarizing information obtained from multiple studies and make inference that is not dependent on any distributional assumption for the study level parameters specifically we assume the study level parameters are unknown fixed parameters and draw inferences about for example the quantiles of this set of parameters using study specific summary statistics this type of problem is known to be quite challenging see hall and miller we

universal filtering via prediction we consider the filtering problem where a finite alphabet individual sequence is corrupted by a discrete memoryless channel and the goal is to causally estimate each sequence component based on the past and present noisy observations we establish a correspondence between the filtering problem and the problem of prediction of individual sequences which leads to the following result given an arbitrary finite set of filters there exists a filter which performs with high probability essentially as well as the best in the set regardless of the underlying noiseless individual sequence we use this relationship between the problems to derive a filter guaranteed of attaining the finite state filterability of any individual sequence by leveraging results from the prediction problem

automatic noise removal signal removal based on general cross validation thresholding in synchrosqueezed domain and its application on earthquake data abstractrecorded seismic signals are often corrupted by noise we have developed an automatic noise attenuation method for single channel seismic data based upon high resolution time frequency analysis synchrosqueezing is a time frequency reassignment method aimed at sharpening a time frequency picture noise can be distinguished from the signal and attenuated more easily in this reassigned domain the threshold level is estimated using a general cross validation approach that does not rely on any prior knowledge about the noise level the efficiency of the thresholding has been improved by adding a preprocessing step based on kurtosis measurement and a postprocessing step based on adaptive hard thresholding the proposed algorithm can either attenuate the noise either white or colored and keep the signal or remove the signal and keep the noise hence it can be used in either normal denoising applications or preprocessing in ambient noise studies we tested the performance of the proposed method on s

on a new generalization of hardy hilbert s inequality and its applications abstract this paper deals with a generalization of the hardy hilbert inequality with best constant factor which involves the function as an application we obtain a new equivalent form of the hardy hilbert inequality

improving the readability of time frequency and time scale representations by the reassignment method in this paper the use of the reassignment method first applied by kodera gendrin and de villedary 1976 to the spectrogram is generalized to any bilinear time frequency or time scale distribution this method creates a modified version of a representation by moving its values away from where they are computed so as to produce a better localization of the signal components we first propose a new formulation of this method followed by a thorough theoretical study of its characteristics its practical use for a large variety of known time frequency and time scale distributions is then addressed finally some experimental results are reported to demonstrate the performance of this method

the robustness of seismic attenuation measurements using fixed and variable window time frequency transforms frequency based methods for measuring seismic attenuation are used commonly in exploration geophysics to measure the spectrum of a nonstationary seismic signal different methods are available including transforms with time windows that are either fixed or systematically varying with the frequency being analyzed we compare four time frequency transforms and show that the choice of a fixed or variable window transform affects the robustness and accuracy of the resulting attenuation measurements for fixed window transforms we use the short time fourier transform and gabor transform the s transform and continuous wavelet transform are analyzed as the variable length transforms first we conduct a synthetic transmission experiment and compare the frequency dependent scattering attenuation to the theoretically predicted values from this procedure we find that variable window transforms reduce the uncertainty and biasof the resulting attenuation estimate specifically at the upper and lower ends of th

applications of the synchrosqueezing transform in seismic time frequency analysis abstracttime frequency representation of seismic signals provides a source of information that is usually hidden in the fourier spectrum the short time fourier transform and the wavelet transform are the principal approaches to simultaneously decompose a signal into time and frequency components known limitations such as trade offs between time and frequency resolution may be overcome by alternative techniques that extract instantaneous modal components empirical mode decomposition aims to decompose a signal into components that are well separated in the time frequency plane allowing the reconstruction of these components on the other hand a recently proposed method called the synchrosqueezing transform sst is an extension of the wavelet transform incorporating elements of empirical mode decomposition and frequency reassignment techniques this new tool produces a well defined time frequency representation allowing the identification of instantaneous frequencies in seismic signals to highlight

on new generalizations of hilbert s inequality and their applications this paper deals with new generalizations of hilbert s inequality and their applications it is shown that results of this paper are significant extensions and improvements of many known results

estimation of unimodal densities without smoothness assumptions the grenander estimator of a decreasing density which is defined as the derivative of the concave envelope of the empirical c d f is known to be a very good estimator of an unknown decreasing density on the half line mathbb r when this density is not assumed to be smooth it is indeed the maximum likelihood estimator and one can get precise upper bounds for its risk when the loss is measured by the mathbb l 1 distance between densities moreover if one restricts oneself to the compact subsets of decreasing densities bounded by h with support on 0 l the risk of this estimator is within a fixed factor of the minimax risk the same is true if one deals with the maximum likelihood estimator for unimodal densities with known mode when the mode is unknown the maximum likelihood estimator does not exist any more we shall provide a general purpose estimator together with a computational algorithm for estimating nonsmooth unimodal densities its risk is the same as the risk of the grenander estimator based on the knowledge of the true mode plus some lower order term it can also cope with small departures from unimodality

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

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

instantaneous spectral bandwidth and dominant frequency with applications to seismic reflection data fourier power spectra are often usefully characterized by average measures in reflection seismology the important average measures are center frequency spectral bandwidth and dominant frequency these quantities have definitions familiar from probability theory center frequency is the spectral mean spectral bandwidth is the standard deviation about that mean and dominant frequency is the square root of the second moment which serves as an estimate of the zero crossing frequency these measures suggest counterparts defined with instantaneous power spectra in place of fourier power spectra so that they are instantaneous in time though they represent averages in frequency intuitively reasonable requirements yield specific forms for these instantaneous quantities that can be computed with familiar complex seismic trace attributes instantaneous center frequency is just instantaneous frequency instantaneous bandwidth is the absolute value of the derivative of the instantaneous amplitude divided by t

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

on estimating mixing densities in discrete exponential family models this paper concerns estimating a mixing density function g and its derivatives based on iid observations from f x f x g d where f x is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers fourier methods are used to derive kernel estimators upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence if f x i 0 e x 1 x for some positive numbers 0 and e then our estimators achieve the optimal rate of convergence log n m for estimating the mth derivative of g under a lipschitz condition of order a m the optimal rate of convergence is almost achieved when x f x 0 e x 1 estimation of the mixing distribution function is also considered

empirical bayes prediction for the multivariate newsvendor loss function motivated by an application in inventory management we consider the multi product newsvendor problem of nding the optimal stocking levels that minimize the total backorder and lost sales costs we focus on a setting where we have a large number of products and observe only noisy estimates of the underlying demand we develop an empirical bayes methodology for predicting stocking levels using data adaptive linear shrinkage strategies which are constructed by minimizing uniformly ecient asymptotic risk estimates in calculating the magnitude and direction of shrinkage our proposed predictive rules incorporate the asymmetric nature of the piecewise linear newsvendor loss function and are shown to be asymptotically optimal using simulated data we study the non asymptotic performance of our method and obtain encouraging results the asymptotic risk estimation method and the proof techniques developed here can also be used to formulate optimal empirical bayes predictive strategies for general piecewise linear and related asymmetric loss functions 1 introduction the newsvendor problem of determining optimal stocking levels is one of the classical problems in the literature on inventory management arrow harris and marschak 1951 choi 2012 we study the traditional newsvendor problem scarf 1959 in the modern big data regime and consider the inventory management problem of a vendor who sells a large number of products the demand distribution of each product is unknown and must be estimated from data our problem is motivated by modern day online retailers who carry hundreds to millions of products the historical sales and projected demand varies greatly across dierent products and the company must decide the stocking quantity of each product we consider a one period setting where based on the observed demand in the previous period the rm must determine the stocking quantity of each product in the next period the observed demand in the previous period provides a basis for estimating the demand in the next period the rm has to balance the tradeos between stocking too much and incurring high inventory cost versus stocking too little and suering lost sales our objective is to develop a data driven policy that minimizes the total inventory and lost sales costs across all products

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

inference for monotone functions under short and long range dependence confidence intervals and new universal limits abstractwe introduce new point wise confidence interval estimates for monotone functions observed with additive dependent noise our methodology applies to both short and long range dependence regimes for the errors the interval estimates are obtained via the method of inversion of certain discrepancy statistics this approach avoids the estimation of nuisance parameters such as the derivative of the unknown function which previous methods are forced to deal with the resulting estimates are therefore more accurate stable and widely applicable in practice under minimal assumptions on the trend and error structure the dependence of the errors especially long range dependence leads to new phenomena where new universal limits based on convex minorant functionals of drifted fractional brownian motion emerge some extensions to uniform confidence bands are also developed supplementary materials for this article are available online

maximum likelihood estimation of smooth monotone and unimodal densities we study the nonparametric estimation of univariate monotone and unimodal densities using the maximum smoothed likelihood approach the monotone estimator is the derivative of the least concave majorant of the distribution corresponding to a kernel estimator we prove that the mapping on distributions with density the derivative of the least concave majorant of is a contraction in all l p norms 1 p and some other distances such as the hellinger and kullback leibler distances the contractivity implies error bounds for monotone density estimation almost the same error bounds hold for unimodal estimation

smoothing under diffeomorphic constraints with homeomorphic splines in this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms this diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time dependent vector field this solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques this method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost in a one dimensional setting incorporating diffeomorphic constraints is equivalent to imposing monotonicity thus as an illustration it is shown that such a monotone spline can be used to make monotone any unconstrained estimator of a regression function and that this monotone smoother inherits the convergence properties of the unconstrained estimator some numerical experiments are proposed to illustrate its finite sample performances and to compare them with another monotone estimator we also provide a two dimensional application on the computation of diffeomorphisms for landmark and image matching

application of non parametric empirical bayes estimation to high dimensional classification we consider the problem of classification using high dimensional features space in a paper by bickel and levina 2004 it is recommended to use naive bayes classifiers that is to treat the features as if they are statistically independent consider now a sparse setup where only a few of the features are informative for classification fan and fan 2008 suggested a variable selection and classification method called fair the fair method improves the design of naive bayes classifiers in sparse setups the improvement is due to reducing the noise in estimating the features means this reduction is since that only the means of a few selected variables should be estimated we also consider the design of naive bayes classifiers we show that a good alternative to variable selection is estimation of the means through a certain non parametric empirical bayes procedure in sparse setups the empirical bayes implicitly performs an efficient variable selection it also adapts very well to non sparse setups and has the advantage of making use of the information from many weakly informative variables which variable selection type of classification procedures give up on using we compare our method with fair and other classification methods in simulation for sparse and non sparse setups and in real data examples involving classification of normal versus malignant tissues based on microarray data

optimal estimation of bacterial growth rates based on a permuted monotone matrix motivated by the problem of estimating the bacterial growth rates for genome assemblies from shotgun metagenomic data we consider the permuted monotone matrix model y theta pi z where y in mathbb r n times p is observed theta in mathbb r n times p is an unknown approximately rank one signal matrix with monotone rows pi in mathbb r p times p is an unknown permutation matrix and z in mathbb r n times p is the noise matrix this paper studies the estimation of the extreme values associated to the signal matrix theta including its first and last columns as well as their difference treating these estimation problems as compound decision problems minimax rate optimal estimators are constructed using the spectral column sorting method numerical experiments through simulated and synthetic microbiome metagenomic data are presented showing the superiority of the proposed methods over the alternatives the methods are illustrated by comparing the growth rates of gut bacteria between inflammatory bowel disease patients and normal controls

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

on hilbert s integral inequality abstract in this paper we generalize hilbert s integral inequality and its equivalent form by introducing three parameterst a andb iff g l2 0 then formula where is the best value the inequality 1 is well known as hilbert s integral inequality and its equivalent form is formula where 2is also the best value cf 1 chap 9 recently hu ke made the following improvement of 1 by introducing a real functionc x formula wherek x 2 0 c t2x 1 t2 dt c x 1 c x c y 0 andf g 0 cf 2 in this paper some generalizations of 1 and 2 are given in the following theorems which are other than those in 2

general maximum likelihood empirical bayes estimation of normal means we propose a general maximum likelihood empirical bayes gmleb method for the estimation of a mean vector based on observations with i i d normal errors we prove that under mild moment conditions on the unknown means the average mean squared error mse of the gmleb is within an infinitesimal fraction of the minimum average mse among all separable estimators which use a single deterministic estimating function on individual observations provided that the risk is of greater order than log n 5 n we also prove that the gmleb is uniformly approximately minimax in regular and weak l p balls when the order of the length normalized norm of the unknown means is between log n k1 n 1 p 2 and n log n k2 simulation experiments demonstrate that the gmleb outperforms the james stein and several state of the art threshold estimators in a wide range of settings without much down side

some stochastics on monotone functions abstract a measure of departures of monotonicity of a given function the l r dip 1 r is introduced our analysis is performed to cover two different situations when the function is known our interest is related to its behavior in a stochastic model however in most cases the knowledge of the function is obtained through a preliminary estimation of the function in both situations the aim focuses in the obtainment of strong consistency results

nonparametric estimation of stage occupation probabilities in a multistage model with current status data multistage models are used to describe individuals or experimental units moving through a succession of stages corresponding to distinct states e g healthy diseased diseased with complications dead the resulting data can be considered to be a form of multivariate survival data containing information about the transition times and the stages occupied traditional survival analysis is the simplest example of a multistage model where individuals begin in an initial stage say alive and move irreversibly to a second stage death in this article we consider general multistage models with a directed tree structure progressive models in which individuals traverse through stages in a possibly non markovian manner we construct nonparametric estimators of stage occupation probabilities and marginal cumulative transition hazards empirical calculations of these quantities are not possible due to the lack of complete data we consider current status information which represents a more severe form of censoring than the commonly used right censoring asymptotic validity of our estimators can be justified using consistency results for nonparametric regression estimators finite sample behavior of our estimators is studied by simulation in which we show that our estimators based on these limited data compare well with those based on complete data we also apply our method to a real life data set arising from a cardiovascular diseases study in taiwan

nonparametric state price density estimation using constrained least squares and the bootstrap the economic theory of option pricing imposes constraints on the structure of call functions and state price densities except in a few polar cases it does not prescribe functional forms this paper proposes a nonparametric estimator of option pricing models which incorporates various restrictions such as monotonicity and convexity within a single least squares procedure the bootstrap is used to produce confidence intervals for the call function and its first two derivatives and to calibrate a residual regression test of shape constraints we apply the techniques to option pricing data on the dax

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

asymptotic properties of maximum likelihood estimates in the mixed poisson model this paper considers the asymptotic behavior of the maximum likelihood estimators mle s of the probabilities of a mixed poisson distribution with a nonparametric mixing distribution the vector of estimated probabilities is shown to converge in probability to the vector of mixed probabilities at rate n 1 2 varepsilon for any varepsilon 0 under a generalized chi 2 distance function it is then shown that any finite set of the mle s has the same joint limiting distribution as does the corresponding set of sample proportions when the support of the mixing distribution g 0 is an infinite set with a known upper bound and g 0 satisfies a certain condition at zero

nonparametric estimation of mixing densities for discrete distributions by a mixture density is meant a density of the form d where is a family of probability densities and is a probability measure on we consider the problem of identifyin

an empirical bayes method for chi squared data in a thought provoking paper efron investigated the merit and limitation of an empirical bayes method to correct selection bias based on tweedie s formula first reported in the study by robbins t

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

the poisson compound decision problem revisited the compound decision problem for a vector of independent poisson random variables with possibly different means has a half century old solution however it appears that the classical solution needs smoothing adjustment we discuss three such adjustments we also present another approach that first transforms the problem into the normal compound decision problem a simulation study shows the effectiveness of the procedures in improving the performance over that of the classical procedure a real data example is also provided the procedures depend on a smoothness parameter that can be selected using a nonstandard cross validation step which is of independent interest finally we mention some asymptotic results

seismic data decomposition into spectral components using regularized nonstationary autoregression summary seismic data can be decomposed into nonstationary spectral components with smoothly variable frequencies and smoothly variable amplitudes to estimate local frequencies i use a nonstationary version of prony s spectral analysis method defined with the help of regularized nonstationary autoregression rnar to estimate local amplitudes of different components i fit their sum to the data using regularized nonstationary regression rnr shaping regularization ensures stability of the estimation process and provides controls on smoothness potential applications of the proposed technique include noise attenuation seismic data compression and seismic data regularization

nonparametric estimation of a regression function consider the regression model y i g x i e i i 1 2 n where x i s denote unordered design variables and g is an unknown function defined on the interval 0 1 assume e i are iid random variables with zero mean and finite variance priestley and chao 1972 and clark 1977 proposed estimators g 2n and g 3n respectively for g in this paper the asymptotic behavior of g 2n and g 3n is studied utilizing the properties of the new estimator g 1n it is shown that g 1n g 2n g 3n are asymptotically equivalent in various senses moreover consistency results are established and rates of uniform convergence obtained for example if e e 3 if g is lipschitz of order 1 and if n is any sequence of constants tending to as n then for all 0 a leqq b 1 n 1 mathord left vphantom 1 3 right kern nulldelimiterspace 3 mathord left vphantom n 1 mathord left vphantom 1 3 right kern nulldelimiterspace 3 beta n log n mathop sup limits a leqq x leqq b g 1n x g x xrightarrow w p 1 0 right kern nulldelimiterspace beta n log n mathop sup limits a leqq x leqq b g 1n x g x xrightarrow w p 1 0 as n finally when g is monotone a strong consistent isotonic estimator g n is considered

on hardy hilbert s integral inequality abstract in this paper by introducing some parameters we give a new generalization of hardy hilbert s integral inequality with a best constant factor that involves the function we also consider its more extended form

on new generalizations of hilbert s inequality abstract by introducing three parameters a b and we give some generalizations of hilbert s integral inequality and its equivalent form with the best constant factors we also consider their associated double series forms

adaptive demixing in poisson mixture models let x 1 x 2 x n be an i i d sample from the poisson mixture distribution p x 1 x 0 s x e s f s ds rates of convergence in mean integrated squared error mise of orthogonal series estimators for the mixing density f supported on a b are studied for the holder class of densities whose r th derivative is lipschitz a the mise converges at the rate log n log log n 2 r for sobolev classes of densities whose r th derivative is square integrable the mise converges at the rate log n log log n 2r the estimator is adaptive over both these classes for the sobolev class a lower bound on the minimax rate of convergence is log n log log n 2r and so the orthogonal polynomial estimator is rate optimal

the limit distribution of the l error of grenander type estimators let f be a nonincreasing function defined on 0 1 under standard regularity conditions we derive the asymptotic distribution of the supremum norm of the difference between f and its grenander type estimator on sub intervals of 0 1 the rate of convergence is found to be of order n log n 1 3 and the limiting distribution to be gumbel

density estimation for the case of supersmooth measurement error abstract the problem is to estimate the probability density of a random variable contaminated by an independent measurement error i explore one of the worst case scenario when the characteristic function of this measurement error decreases exponentially and thus optimal estimators converge only with logarithmic rate the particular example of such measurement error is any random variable contaminated by normal cauchy or another stable random variable for this setting and circular data i suggest an asymptotically efficient data driven estimator that is adaptive to both smoothness of estimated density and distribution of measurement error moreover this estimator is universal in sense that its derivatives and integral are sharp estimators of the corresponding derivatives and the cumulative distribution function and these estimators are sharp both globally and pointwise for the case of small sample sizes i suggest a modified estimator that mimics an optimal linear pseudoestimator i explore this est

a new algorithm for multicomponent signals analysis based on synchrosqueezing with an application to signal sampling and denoising in this paper we address the problem of the retrieval of the components from a multicomponent signal using ideas from the synchrosqueezing framework the emphasis is on the wavelet choice and we propose a novel algorithm based first on the detection of components followed by their reconstruction simulations illustrate how the proposed procedure compares with the empirical mode decomposition and other related methods in terms of mode mixing we conclude the paper by studying the sensitivity of the proposed technique to sampling and an application to signal denoising

wavelet deconvolution this paper studies the issue of optimal deconvolution density estimation using wavelets the approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation we explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients minimax rates of convergence under the integrated square loss are studied over besov classes b sub spl sigma pq of functions for both ordinary smooth and supersmooth convolution kernels the minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels it is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the besov spaces with p 2 when the convolution kernel is ordinary smooth and super smooth if the convolution kernel is ordinary smooth then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed

the root unroot algorithm for density estimation as implemented via wavelet block thresholding we propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem this regression problem is created by binning the original observations into many small size bins and by then applying a suitable form of root transformation to the binned data counts in principle many common nonparametric regression estimators could then be applied to the transformed data we propose use of a wavelet block thresholding estimator in this paper finally the estimated regression function is un rooted by squaring and normalizing the density estimation procedure achieves simultaneously three objectives computational efficiency adaptivity and spatial adaptivity a numerical example and a practical data example are discussed to illustrate and explain the use of this procedure theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the besov classes the estimator also automatically adapts to the local smoothness of the underlying function and attains the local adaptive minimax rate for estimating functions at a point there are three key steps in the technical argument poissonization quantile coupling and oracle risk bound for block thresholding in the non gaussian setting some of the technical results may be of independent interest

some asymptotics for multimodality tests based on kernel density estimates a test due to b w silverman for modality of a probability density is based on counting modes of a kernel density estimator and the idea of critical smoothing an asymptotic formula is given for the expected number of modes this together with other methods establishes the rate of convergence of the critically smoothed bandwidth these ideas are extended to provide insight concerning the behaviour of the test based on bootstrap critical values

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

monotone regression estimates for grouped observations the maximum likelihood estimator of a nondecreasing regression function with normally distributed errors has been considered in the literature its asymptotic distribution at a point is related to a solution of the heat equation and its rate of convergence to the underlying regression function is of order n 1 3 this estimator can be modified by grouping adjacent observations and then isotonizing the corresponding means it is shown that the resulting estimator has an asymptotic normal distribution for certain group sizes and its rate of convergence is of order n 215 the results of a simulation study for small sample sizes are presented and grouping procedures are discussed

nonparametric empirical bayes and compound decision approaches to estimation of a high dimensional vector of normal means we consider the classical problem of estimating a vector 1 n based on independent observations y i n i 1 i 1 n suppose i i 1 n are independent realizations from a completely unknown g we suggest an easily computed estimator such that the ratio of its risk e 2 with that of the bayes procedure approaches 1 a related compound decision result is also obtained our asymptotics is of a triangular array that is we allow the distribution g to depend on n thus our theoretical asymptotic results are also meaningful in situations where the vector is sparse and the proportion of zero coordinates approaches 1 we demonstrate the performance of our estimator in simulations emphasizing sparse setups in moderately sparse situations our procedure performs very well compared to known procedures tailored for sparse setups it also adapts well to nonsparse situations

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

on the extended hardy hilbert s inequality abstract in this paper by introducing three parameters a b and and estimating the weight coefficient we give a generalization of the extended hardy hilbert s inequality with a best possible constant factor involving the function we also consider its equivalent inequality and the associated double series form

a computational framework for multivariate convex regression and its variants we study the nonparametric least squares estimator lse of a multivariate convex regression function the lse given as the solution to a quadratic program with o n2 linear constraints n being t

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

admissible estimation dirichlet principles and recurrence of birth death chains on ℤ p we establish a connection between admissible simultaneous estimation and recurrence of reversible markov chains on specifically to each generalized bayes estimator of the mean of a vector of p independent poisson variables for a weighted quadratic loss we associate a variational problem and a reversible birth and death chain on the variational problem is closely related to the dirichlet principle for reversible chains studied recently by griffeath liggett and lyons under side conditions admissibility of the estimator is equivalent to zero infimal energy in the variational problem and to recurrence of the markov chain this yields analytic and probabilistic criteria for inadmissibility which are applied to establish a broad class of results and previous conjectures