on extension of isometries between unit spheres of l γ type space and a banach space e abstract in this paper we study the isometric extension problem between the unit spheres of l type space and a banach space e we prove that every surjective isometry between the unit spheres s l and s e can be extended to be a linear isometry of l onto e moreover we study this problem between c and a banach space and also get an affirmative answer

on the sampling distribution of resubstitution and leave one out error estimators for linear classifiers error estimation is a problem of high current interest in many areas of application this paper concerns the classical problem of determining the performance of error estimators in small sample settings under a gaussianity parametric assumption we provide here for the first time the exact sampling distribution of the resubstitution and leave one out error estimators for linear discriminant analysis lda in the univariate case which is valid for any sample size and combination of parameters including unequal variances and sample sizes for each class in the multivariate case we provide a quasi binomial approximation to the distribution of both the resubstitution and leave one out error estimators for lda under a common but otherwise arbitrary class covariance matrix which is assumed to be known in the design of the lda discriminant we provide numerical examples using both synthetic and real data that indicate that these approximations are accurate provided that lda classification error is not too large

bayesian regression with network prior optimal bayesian filtering perspective the recently introduced intrinsically bayesian robust filter ibrf provides fully optimal filtering relative to a prior distribution over an uncertainty class of joint random process models whereas formerly the theory was limited to model constrained bayesian robust filters for which optimization was limited to the filters that are optimal for models in the uncertainty class this paper extends the ibrf theory to the situation where there are both a prior on the uncertainty class and sample data the result is optimal bayesian filtering obf where optimality is relative to the posterior distribution derived from the prior and the data the ibrf theories for effective characteristics and canonical expansions extend to the obf setting a salient focus of the present study is to demonstrate the advantages of bayesian regression within the obf setting over the classical bayesian approach in the context of linear gaussian models

instrumental variable estimation in functional linear models in an increasing number of empirical studies the dimensionality measured e g as the size of the parameter space of interest can be very large two instances of large dimensional models are the linear regression with a large number of covariates and the estimation of a regression function with many instrumental variables an appropriate setting to analyze high dimensional problems is provided by a functional linear model in which the covariates belong to hilbert spaces this paper considers the case where covariates are endogenous and assumes the existence of instrumental variables that are functional as well the paper shows that estimating the regression function is a linear ill posed inverse problem with a known but data dependent operator the first contribution is to analyze the rate of convergence of the penalized least squares estimator based on the result we discuss the notion of instrument strength in the high dimensional setting we also discuss a generalized version of the estimator when the problem is premultiplied by an instrument dependent operator this extends the technology of generalized method of moments to high dimensional functional data a central limit theorem is also established on the inner product of the estimator the studied estimators are easy and fast to implement and the finite sample performance is discussed through simulations and an application to the impact of age specific fertility rate curves on yearly economic growth in the united kingdom

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part i representation in recent years biomedicine has been faced with difficult high throughput small sample classification problems in such settings classifier error estimation becomes a critical issue because training and testing must be done on the same data a recently proposed error estimator places the problem in a signal estimation framework in the presence of uncertainty permitting a rigorous solution optimal in a minimum mean square error sense the uncertainty in this model is relative to the parameters of the feature label distributions resulting in a bayesian approach to error estimation closed form solutions are available for two important problems discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors in this work part i of a two part study we introduce the theoretical mean square error mse conditioned on the observed sample of any estimate of the classifier error including the bayesian error estimator for both bayesian models thus bayesian error estimation has a unique advantage in that its mathematical framework naturally gives rise to a practical expected measure of performance given an observed sample in part ii of the study we examine consistency of the error estimator demonstrate various mse properties and apply the conditional mse to censored sampling

analytic representation of bayes labeling and bayes clustering operators for random labeled point processes clustering algorithms typically group points based on some similarity criterion but without reference to an underlying random process to make clustering algorithms rigorously predictive in fact there exists a probabilistic theory of clustering in the context of random labeled point sets in which clustering error is defined in terms of the process in the present paper given an underlying point process we develop a general analytic procedure for finding an optimal clustering operator the bayes clusterer that corresponds to the bayes classifier in classification theory we provide detailed solutions under gaussian models owing to computational complexity we also develop approximations of the bayes clusterer

bayesian minimum mean square error estimation for classification error part i definition and the bayesian mmse error estimator for discrete classification with the advent of high throughput genomic and proteomic technologies in conjunction with the difficulty in obtaining even moderately sized samples small sample classifier design has become a major issue in the biological and medical communities with small samples training data error estimation becomes mandatory yet none of the popular error estimation techniques have been rigorously designed based on statistical inference and optimization in this investigation we place classifier error estimation into the framework of optimal mean square error mse signal estimation in the presence of uncertainty which results in a bayesian approach to error estimation based on a parameterized family of feature label distributions with the prior distribution of the parameters governing the choice of feature label distribution these bayesian error estimators are optimal when averaged over a given family of distributions unbiased when averaged over a given family and all samples and analytically address a trade off between robustness modeling assumptions and accuracy minimum mean square error in this paper part i of a two part study we define the minimum mean square error mmse error estimator discuss its basic properties provide closed form analytic estimator representation for discrete classifiers with both non informative and informative prior distributions and examine the performance and robustness of the mmse error estimator via simulations in part ii of this paper in this same issue of ieee transactions on signal processing we address all of these issues in particular closed form representation for linear classification in the gaussian model with known and unknown covariance matrices for both the discrete and gaussian cases the mmse error estimator has especially good performance for distributions having moderate true errors

smoothing splines estimators for functional linear regression the paper considers functional linear regression where scalar responses y1 yn are modeled in dependence of random functions x1 xn we propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty theoretical analysis concentrates on the error in an out ofsample prediction of the response for a new random function xn 1 it is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors we then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves for the case of models with errors in variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves the methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day

a comparison of asymptotic error rate expansions for the sample linear discriminant function abstract several asymptotic expansions for approximating the expected or unconditional probability of misclassification for the sample linear discriminant function are compared for accuracy in terms of yielding the smallest mean absolute deviation from the exact value for 104 population configurations the actual expected probabilities of misclassification are found via monte carlo simulation a simple and relatively obscure asymptotic expansion derived by raudys tech cybern 4 168 174 1972 is found to yield better approximation than the well known asymptotic expansions

minimax mse estimation of deterministic parameters with noise covariance uncertainties in this paper a minimax mean squared error mse estimator is developed for estimating an unknown deterministic parameter vector in a linear model subject to noise covariance uncertainties the estimator is designed to minimize the worst case mse across all norm bounded parameter vectors and all noise covariance matrices in a given region of uncertainty the minimax estimator is shown to have the same form as the estimator that minimizes the worst case mse over all norm bounded vectors for a least favorable choice of the noise covariance matrix an example demonstrating the performance advantage of the minimax mse approach over the least squares and weighted least squares methods is presented

discrete optimal bayesian classification with error conditioned sequential sampling when in possession of prior knowledge concerning the feature label distribution in particular when it is known that the feature label distribution belongs to an uncertainty class of distributions governed by a prior distribution this prior knowledge can be used in conjunction with the training data to construct the optimal bayesian classifier obc whose performance is on average optimal among all classifiers relative to the posterior distribution derived from the prior distribution and the data typically in classification theory it is assumed that sampling is performed randomly in accordance with the prior probabilities on the classes and this has heretofore been true in the case of obc in the present paper we propose to forego random sampling and utilize the prior knowledge and previously collected data to determine which class to sample from at each step of the sampling specifically we choose to sample from the class that leads to the smallest expected classification error with the addition of the new sample point we demonstrate the superiority of the resulting nonrandom sampling procedure to random sampling on both synthetic data and data generated from known biological pathways highlightsa sampling algorithm for training the optimal bayesian classifier is introduced the algorithm works based on minimization of the expected error on the uncertainty class of prior knowledge using a zipf model we show that our sampling algorithm leads to a less true error on average than random sampling our algorithm shows robustness even in case when prior knowledge drifts away from true distributions an example on data from p53 network shows that our method works well on from real pathway data as well

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

estimation of risk neutral densities using positive convolution approximation this paper proposes a new nonparametric method for estimating the conditional risk neutral density rnd from a cross section of option prices the idea of the method is to t option prices by nding the optimal density in a special admissible set the admissible set consists of functions each of which may be represented as a convolution of a positive kernel with another density the method is termed the positive convolution approximation pca the important properties of pca are that it 1 is completely agnostic about the data generating process 2 controls against over tting while allowing for small samples 3 always produces arbitragefree estimators and 4 is computationally simple in a monte carlo experiment pca is compared to several popular methods mixtures of lognormals with one two and three lognormals hermite polynomials two regularization methods for the rnd and for implied volatilities and sigma shape polynomials pca is found to be a promising alternative to the competitors c 2003 elsevier b v all rights reserved

optimal robust classifiers qualitatively a filter is said to be robust if its performance degradation is acceptable for distributions close to the one for which it is optimal that is the one for which it has been designed this paper adapts the signal processing theory of optimal robust filters to classifiers the distribution class conditional distributions to which the classifier is to be applied is parameterized by a state vector and the principle issue is to choose a design state that is optimal in comparison to all other states relative to some measure of robustness a minimax robust classifier is one whose worst performance over all states is better than the worst performances of the other classifiers defined at the other states a bayesian robust classifier is one whose expected performance is better than the expected performances of the other classifiers the state corresponding to the bayesian robust classifier is called the maximally robust state minimax robust classifiers tend to give too much weight to states for which classification is very difficult and therefore our effort is focused on bayesian robust classifiers whereas the signal processing theory of robust filtering concentrates on design with full distributional knowledge and a fixed number of observation variables features design via training from sample data and feature selection are so important for classification that robustness optimality must be considered from these perspectives in particular for small samples in this context for a given sample size we will be concerned with the maximally robust state feature pair all definitions are independent of the classification rule however applications are only considered for linear and quadratic discriminant analysis for which there are parametric forms for the optimal discriminants

adaptive functional linear regression we consider the estimation of the slope function in functional linear regression where scalar responses are modeled in dependence of random functions cardot and johannes j multivariate anal 101 2010 395 408 have shown that a thresholded projection estimator can attain up to a constant minimax rates of convergence in a general framework which allows us to cover the prediction problem with respect to the mean squared prediction error as well as the estimation of the slope function and its derivatives this estimation procedure however requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor as this information is usually inaccessible in practice we investigate a fully data driven choice of the tuning parameter which combines model selection and lepski s method it is inspired by the recent work of goldenshluger and lepski ann statist 39 2011 1608 1632 the tuning parameter is selected as minimizer of a stochastic penalized contrast function imitating lepski s method among a random collection of admissible values this choice of the tuning parameter depends only on the data and we show that within the general framework the resulting data driven thresholded projection estimator can attain minimaxrates up to a constant over a variety of classes of slope functions and covariance operators the results are illustrated considering different configurations which cover in particular the prediction problem as well as the estimation of the slope and its derivatives a simulation study shows the reasonable performance of the fully data driven estimation procedure

linear minimax regret estimation of deterministic parameters with bounded data uncertainties we develop a new linear estimator for estimating an unknown parameter vector x in a linear model in the presence of bounded data uncertainties the estimator is designed to minimize the worst case regret over all bounded data vectors namely the worst case difference between the mean squared error mse attainable using a linear estimator that does not know the true parameters x and the optimal mse attained using a linear estimator that knows x we demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least squares estimator as well as several other least squares alternatives

analytic study of performance of error estimators for linear discriminant analysis we derive double asymptotic analytical expressions for the first moments second moments and cross moments with the actual error for the resubstitution and leave one out error estimators in the case of linear discriminant analysis in the multivariate gaussian model under the assumption of a common known covariance matrix and a fixed mahalanobis distance as dimensionality approaches infinity sample sizes for the two classes need not be the same they are only assumed to reach a fixed but arbitrary asymptotic ratio with the dimensionality from the asymptotic moment representations we directly obtain double asymptotic expressions for the bias variance and rms of the error estimators the asymptotic expressions presented here generally provide good small sample approximations as demonstrated via numerical experiments the applicability of the theoretical results is illustrated by finding the minimum sample size to bound the rms in gene expression classification

characterizations on mazur ulam theorem abstract we consider the area preserving 2 isometries in linear 2 normed spaces in chu 2007 10 the mazur ulam theorem was proved in linear 2 normed spaces with the assumption of the collinearity preserving in this article we prove the mazur ulam theorem for the interior preserving mappings in linear 2 normed spaces recently amyari sadeghi 16 in non archimedean 2 normed spaces the mazur ulam theorem was proved under the condition of strict convexity in this paper we also prove the theorem on non archimedean 2 normed spaces over a linear ordered non archimedean field without the strict convexity assumption

optimal mean square error calibration of classifier error estimators under bayesian models a recently proposed bayesian modeling framework for classification facilitates both the analysis and optimization of error estimation performance the bayesian error estimator is then defined to have optimal mean square error performance but in many situations closed form representations are unavailable and approximations may not be feasible to address this we present a method to optimally calibrate arbitrary error estimators for minimum mean square error performance within a supposed bayesian framework assuming a fixed sample size classification rule and error estimation rule as well as a fixed bayesian model the calibration is done by first computing a calibration function that maps error estimates to their optimally calibrated values off line once found this calibration function may be easily applied to error estimates on the fly whenever the assumptions apply we demonstrate that calibrated error estimators offer significant improvement in performance relative to classical error estimators under bayesian models with both linear and non linear classification rules

the aleksandrov problem in linear 2 normed spaces we introduce the concept of 2 isometry which is suitable to represent the notion of area preserving mappings in linear 2 normed spaces and then we obtain some results for the aleksandrov problem in linear 2 normed spaces

methodology and convergence rates for functional linear regression in functional linear regression the slope parameter is a function therefore in a nonparametric context it is determined by an infinite number of unknowns its estimation involves solving an ill posed problem and has points of contact with a range of methodologies including statistical smoothing and deconvolution the standard approach to estimating the slope function is based explicitly on functional principal components analysis and consequently on spectral decomposition in terms of eigenvalues and eigen functions we discuss this approach in detail and show that in certain circumstances optimal convergence rates are achieved by the pca technique an alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view

characterizations on isometries in linear n normed spaces abstract let f x y be a weak n isometry in linear n normed spaces in this paper we prove that if f satisfies some conditions then f is an n isometry

properties of isometric mappings abstract some relations between isometry and linearity are examined in particular generalizations of the mazur ulam theorem and open problems are discussed

robust control of markov decision processes with uncertain transition matrices optimal solutions to markov decision problems may be very sensitive with respect to the state transition probabilities in many practical problems the estimation of these probabilities is far from accurate hence estimation errors are limiting factors in applying markov decision processes to real world problems we consider a robust control problem for a finite state finite action markov decision process where uncertainty on the transition matrices is described in terms of possibly nonconvex sets we show that perfect duality holds for this problem and that as a consequence it can be solved with a variant of the classical dynamic programming algorithm the robust dynamic programming algorithm we show that a particular choice of the uncertainty sets involving likelihood regions or entropy bounds leads to both a statistically accurate representation of uncertainty and a complexity of the robust recursion that is almost the same as that of the classical recursion hence robustness can be added at practically no extra computing cost we derive similar results for other uncertainty sets including one with a finite number of possible values for the transition matrices we describe in a practical path planning example the benefits of using a robust strategy instead of the classical optimal strategy even if the uncertainty level is only crudely guessed the robust strategy yields a much better worst case expected travel time

inequalities for distances between points and distance preserving mappings abstract in this paper we generalize the short diagonals lemma by proving a new inequality for distances between six points moreover we apply this inequality to a partial solution to the aleksandrov rassias problem

nonparametric instrumental variables estimation of a quantile regression model we consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression error conditional on an instrumental variable to be zero the resulting estimating equation is a nonlinear integral equation of the first kind which generates an ill posed inverse problem the integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically we show that the estimator is mean square consistent derive its rate of convergence in probability and give conditions under which this rate is optimal in a minimax sense the results of monte carlo experiments show that the estimator behaves well in finite samples

functional linear regression with functional response abstract in this paper we develop new estimation results for functional regressions where both the regressor z t and the response y t are functions of hilbert spaces indexed by the time or a spatial location the model can be thought as a generalization of the multivariate regression where the regression coefficient is now an unknown operator we propose to estimate the operator by tikhonov regularization which amounts to apply a penalty on the l 2 norm of we derive the rate of convergence of the mean square error the asymptotic distribution of the estimator and develop tests on as trajectories are often not fully observed we consider the scenario where the data become more and more frequent infill asymptotics we also address the case where z is endogenous and instrumental variables are used to estimate an application to the electricity consumption completes the paper

on the prediction of stationary functional time series this article addresses the prediction of stationary functional time series existing contributions to this problem have largely focused on the special case of first order functional autoregressive processes because of their technical tractability and the current lack of advanced functional time series methodology it is shown here how standard multivariate prediction techniques can be used in this context the connection between functional and multivariate predictions is made precise for the important case of vector and functional autoregressions the proposed method is easy to implement making use of existing statistical software packages and may therefore be attractive to a broader possibly nonacademic audience its practical applicability is enhanced through the introduction of a novel functional final prediction error model selection criterion that allows for an automatic determination of the lag structure and the dimensionality of the model the usefulness of the proposed methodology is demonst

the rate distortion function on classes of sources determined by spectral capacities the quantity sup a in cal r r a d is considered where cal r is a class of homogeneous n parameter sources and r a d denotes the single letter mean square error mse rate distortion function for the individual source a in particular the case in which the class cal r is specified in terms of spectral information is treated for general classes of spectral measures whose upper measures are capacities in the sense of choquet alternating of order two this type of class includes many common models for spectral uncertainty such as mixture models spectral band models and neighborhoods generated by variation and prohorov metrics it is shown that each such class contains a worst case source whose rate distortion function achieves the supremum over the class for each value of distortion this source is characterized as having a spectral density that is a derivative in the sense of huber and strassen of the upper spectral measure with respect to lebesgue measure on pi pi n moreover it is shown that the spectral measure of the worst case source is closest in a sense defined by directed divergence to lebesgue measure which corresponds to a memoryless source numerical results are presented for the particular case in which the source spectral measure is a mixture of a gauss markov spectrum and an unknown contaminating component

robust competitive estimation with signal and noise covariance uncertainties robust estimation of a random vector in a linear model in the presence of model uncertainties has been studied in several recent works while previous methods considered the case in which the uncertainty is in the signal covariance and possibly the model matrix but the noise covariance is assumed to be completely specified here we extend the results to the case where the noise statistics may also be subjected to uncertainties we propose several different approaches to robust estimation which differ in their assumptions on the given statistics in the first method we assume that the model matrix and both the signal and the noise covariance matrices are uncertain and develop a minimax mean squared error mse estimator that minimizes the worst case mse in the region of uncertainty the second strategy assumes that the model matrix is given and tries to uniformly approach the performance of the linear minimum mse estimator that knows the signal and noise covariances by minimizing a worst case regret measure the regret is defined as the difference or ratio between the mse attainable using a linear estimator ignorant of the signal and noise covariances and the minimum mse possible when the statistics are known as we show earlier solutions follow directly from our more general results however the approach taken here in developing the robust estimators is considerably simpler than previous methods

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

nonlinear estimation for a class of systems this paper considers nonlinear estimation problems for classes of models and employs relative entropy to describe the uncertainty classes two optimization problems are formulated on general banach spaces and their solutions are sought 1 when the transition probability between the signal to be estimated x and the measurement y or stochastic kernel is unknown and 2 when the joint probability induced by the random variables rvs x y is unknown for both problems the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution the results include existence of the optimal measures using weak convergence techniques and properties associated with the estimate of the true distribution classical examples are chosen to illustrate the applicability of the results

robust deconvolution of deterministic and random signals we consider the problem of designing an estimation filter to recover a signal x n convolved with a linear time invariant lti filter h n and corrupted by additive noise our development treats the case in which the signal x n is deterministic and the case in which it is a stationary random process both formulations take advantage of some a priori knowledge on the class of underlying signals in the deterministic setting the signal is assumed to have bounded weighted energy in the stochastic setting the power spectra of the signal and noise are bounded at each frequency the difficulty encountered in these estimation problems is that the mean squared error mse at the output of the estimation filter depends on the problem unknowns and therefore cannot be minimized beginning with the deterministic setting we develop a minimax mse estimation filter that minimizes the worst case point wise mse between the true signal x n and the estimated signal over the class of bounded norm inputs we then establish that the mse at the output of the minimax mse filter is smaller than the mse at the output of the conventional inverse filter for all admissible signals next we treat the stochastic scenario for which we propose a minimax regret estimation filter to deal with the power spectrum uncertainties this filter is designed to minimize the worst case difference between the mse in the presence of power spectrum uncertainties and the mse of the wiener filter that knows the correct power spectra the minimax regret filter takes the entire uncertainty interval into account and as demonstrated through an example can often lead to improved performance over traditional minimax mse approaches for this problem

bayesian minimum mean square error estimation for classification error part ii linear classification of gaussian models in this paper part ii of a two part study we derive a closed form analytic representation of the bayesian minimum mean square error mmse error estimator for linear classification assuming gaussian models this is presented in a general framework permitting a structure on the covariance matrices and a very flexible class of prior parameter distributions with four free parameters closed form solutions are provided for known scaled identity and arbitrary covariance matrices we examine performance in small sample settings via simulations on both synthetic and real genomic data and demonstrate the robustness of these error estimators to false gaussian modeling assumptions by applying them to johnson distributions

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

on the generalizations of the mazur ulam isometric theorem abstract this paper contains several generalizations of the mazur ulam isometric theorem in f spaces which are not assumed to be locally bounded let x and y be two real f spaces and let x be locally pseudoconvex or midpoint bounded assume that a operator t maps x onto y in a locally 1 2i isometric manner for all i 0 n then t is affine in addition we give the sufficient conditions of a mapping between two topological vector spaces being affine

optimal penalized function on function regression under a reproducing kernel hilbert space framework many scientific studies collect data where the response and predictor variables are both functions of time location or some other covariate understanding the relationship between these functiona

a contribution to the aleksandrov conservative distance problem in two dimensions abstract let e be a two dimensional real normed space in this paper we show that if the unit circle of e does not contain any line segment such that the distance between its endpoints is greater than 1 then every transformation e e which preserves the unit distance is automatically an affine isometry in particular this condition is satisfied when the norm is strictly convex

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part ii consistency and performance analysis in part i of a two part study on the mse performance of bayesian error estimation we have derived analytical expressions for mse conditioned on the sample for bayesian error estimators and arbitrary error estimators in two bayesian models discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors here in part ii we examine the consistency of bayesian error estimation and provide several simulation studies that illustrate the concept of conditional mse and how it may be used in practice a salient application is censored sampling where sample points are collected one at a time until the conditional mse reaches a stopping criterion

a fixed point theorem in n banach spaces and ulam stability abstract motivated by some issues in ulam stability we prove a fixed point theorem for operators acting on some classes of functions with values in n banach spaces we also present applications of it to ulam stability of eigenvectors and some functional and difference equations

an inequality for distances between 2n points and the aleksandrov rassias problem in this paper we introduce an inequality for distances between every two points among the given 2n points and apply this inequality to a partial solution of the aleksandrov rassias problem which was first posed by th m rassias

characterizations on 2 isometries we consider 2 isometries weak 2 isometries and 2 continuous mappings and investigate the relation between three mappings in linear 2 normed spaces also we prove that the riesz theorem holds when x is a linear 2 normed spaces

on the aleksandrov problem in linear n normed spaces the aim of this article is to generalize the aleksandrov problem to the case of linear n normed spaces

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

mmse bounds for additive noise channels under kullback leibler divergence constraints on the input distribution upper and lower bounds on the minimum mean square error for additive noise channels are derived when the input distribution is constrained to be close to a gaussian reference distribution in terms of the kullback leibler divergence the upper bound is tight and is attained by a gaussian distribution whose mean is identical to that of the reference distribution and whose covariance matrix is defined implicitly via a system of non linear equations the estimator that attains the upper bound is identified as a minimax optimal estimator that is robust against deviations from the assumed prior the lower bound provides an alternative to well known inequalities in estimation and information theory such as the cramer rao lower bound stam s inequality or the entropy power inequality that is potentially tighter and defined for a larger class of input distributions several examples of applications in signal processing and information theory illustrate the usefulness of the proposed bounds in practice

mappings of conservative distances and the mazur ulam theorem abstract let x and y be two real hilbert spaces with the dimension of x greater than 1 several cases about the aleksandrov rassias problem for t x y preserving two or three distances are presented and geometric interpretations of these cases are also given

intrinsically bayesian robust kalman filter an innovation process approach in many contemporary engineering problems model uncertainty is inherent because accurate system identification is virtually impossible owing to system complexity or lack of data on account of availability time or cost the situation can be treated by assuming that the true model belongs to an uncertainty class of models in this context an intrinsically bayesian robust ibr filter is one that is optimal relative to the cost function in the classical sense and the prior distribution over the uncertainty class in the bayesian sense ibr filters have previously been found for both wiener and granulometric morphological filtering in this paper we derive the ibr kalman filter that performs optimally relative to an uncertainty class of state space models introducing the notion of bayesian innovation process and the bayesian orthogonality principle we show how the problem of designing an ibr kalman filter can be reduced to a recursive system similar to the classical kalman recursive equations except with effective counterparts such as the effective kalman gain matrix after deriving the recursive ibr kalman equations for discrete time we use the limiting method to obtain the ibr kalman bucy equations for continuous time finally we demonstrate the utility of the proposed framework for two real world problems sensor networks and gene regulatory network inference

moments and root mean square error of the bayesian mmse estimator of classification error in the gaussian model the most important aspect of any classifier is its error rate because this quantifies its predictive capacity thus the accuracy of error estimation is critical error estimation is problematic in small sample classifier design because the error must be estimated using the same data from which the classifier has been designed use of prior knowledge in the form of a prior distribution on an uncertainty class of feature label distributions to which the true but unknown feature distribution belongs can facilitate accurate error estimation in the mean square sense in circumstances where accurate completely model free error estimation is impossible this paper provides analytic asymptotically exact finite sample approximations for various performance metrics of the resulting bayesian minimum mean square error mmse error estimator in the case of linear discriminant analysis lda in the multivariate gaussian model these performance metrics include the first second and cross moments of the bayesian mmse error estimator with the true error of lda and therefore the root mean square rms error of the estimator we lay down the theoretical groundwork for kolmogorov double asymptotics in a bayesian setting which enables us to derive asymptotic expressions of the desired performance metrics from these we produce analytic finite sample approximations and demonstrate their accuracy via numerical examples various examples illustrate the behavior of these approximations and their use in determining the necessary sample size to achieve a desired rms the supplementary material contains derivations for some equations and added figures highlightsthe bayesian minimum mean square error bmmse error estimator of lda is studied the first second and cross moments of the bmmse estimator with true error are considered both conditional and unconditional performance metrics are analyzed for the first time the kolmogorov double asymptotic is used in a bayesian setting asymptotically exact finite sample approximations of performance metrics are derived

mappings of conservative distances in linear n normed spaces abstract the aim of this article is to generalize the mazur ulam theorem to the case of linear n normed spaces

bandwidth selection in kernel density estimation oracle inequalities and adaptive minimax optimality we address the problem of density estimation with mathbb l s loss by selection of kernel estimators we develop a selection procedure and derive corresponding mathbb l s risk oracle inequalities it is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic nikol skii classes the main technical tools used in our derivations are uniform bounds on the mathbb l s norms of empirical processes developed recently by goldenshluger and lepski ann probab 2011 to appear

the reliability of estimated confidence intervals for classification error rates when only a single sample is available error estimation accuracy is the salient issue regarding the validity of a classifier model when samples are small training data based error estimates tend to suffer from inaccuracy and quantification of error estimation accuracy is difficult numerous methods have been proposed for estimating confidence intervals for the true error based on the estimated error this paper surveys proposed methods and quantifies their performance monte carlo methods are used to obtain accurate estimates of the true confidence intervals and compare these to the intervals estimated from samples we consider different error estimators and several proposed confidence bound estimators both synthetic and real genomic data are employed our simulations show the majority of the confidence intervals methods have poor performance because of the difference of shape between true and estimated intervals according to our results the best estimation strategy is to use the 10 time 10 fold cross validation with a confidence interval based on the standard deviation

tests for error correlation in the functional linear model in this chapter we consider two tests for error correlation in the fully functional linear model which we call methods i and ii they complement the tools described in section 8 6 and the graphical goodness of fit checks used in chapter 9 to construct the test statistics finite dimensional residuals are computed in two different ways and then their autocorrelations are suitably defined from these autocorrelation matrices two quadratic forms are constructed whose limiting distribution are chi squared with known numbers of degrees of freedom different for the two forms the test statistics can be relatively easily computed using the r package fda

on robust wiener signal estimation a minimax formulation is considered for the problem of designing robust linear causal estimators of continuous time signals in noise it is shown that the minimax estimation problem can generally be reduced to finding a least favorable pair of spectra in particular robust predictors filters and finite lag smoothers are shown to be just the optimal predictors filters and smoothers respectively for a least favorable pair this result can often greatly simplify the design of robust estimators

optimal classifiers with minimum expected error within a bayesian framework part ii properties and performance analysis in part i of this two part study we introduced a new optimal bayesian classification methodology that utilizes the same modeling framework proposed in bayesian minimum mean square error mmse error estimation optimal bayesian classification thus completes a bayesian theory of classification where both the classifier error and our estimate of the error may be simultaneously optimized and studied probabilistically within the assumed model having developed optimal bayesian classifiers in discrete and gaussian models in part i here we explore properties of optimal bayesian classifiers in particular invariance to invertible transformations convergence to the bayes classifier and a connection to bayesian robust classifiers we also explicitly derive optimal bayesian classifiers with non informative priors and explore relationships to linear and quadratic discriminant analysis lda and qda which may be viewed as plug in rules under gaussian modeling assumptions finally we present several simulations addressing the robustness of optimal bayesian classifiers to false modeling assumptions companion website http gsp tamu edu publications supplementary dalton12a

pricing and hedging derivative securities with neural networks and a homogeneity hint abstract we estimate a generalized option pricing formula that has a functional shape similar to the usual black scholes formula by a feedforward neural network model this functional shape is obtained when the option pricing function is homogeneous of degree one with respect to the underlying asset price st and the strike price k we show that pricing accuracy gains can be made by exploiting this generalized black scholes shape instead of setting up a learning network mapping the ratio st k and the time to maturity directly into the derivative price we break down the pricing function into two parts one controlled by the ratio st k the other one by a function of time to maturity the results indicate that the homogeneity hint always reduces the out of sample mean squared prediction error compared with a feedforward neural network with no hint both feedforward network models with and without the hint provide similar delta hedging errors that are small relative to the hedging performance of the black scholes model however the model with hint produces a more stable hedging performance

robust matched filters for optical receivers the problem of designing optical receivers that are robust against uncertainty in the statistics of the observation process in photodetection is investigated in particular a modification in the design of the postdetection matched filter is proposed to account for possible uncertainty in the rate function of the incident light the rate of the dark current and in the statistics of the additive noise present at the input to the optical receiver this design is based on a game theoretic approach in which a filter is sought that has the maximum worst case output signal to noise ratio possible over the class of allowable statistics that is the design criterion is maximin signal to noise ratio a general characterization of maximin robust matched filters for observed poisson processes is presented in this context and specific solutions for several useful uncertainty models are obtained numerical results are presented for a specific example to illustrate the performance of the proposed technique

robust hypothesis testing for bounded classes of probability densities corresp least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds the results are shown to extend huber s solution for the epsilon contaminated classes which is contained as a special case a minimum distance property of the least favorable density pair is also made explicit

weakly dependent functional data functional data often arise from measurements on fine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals for example days the functions thus obtained form a functional time series and the central issue in the analysis of such data consists in taking into account the temporal dependence of these functional observations examples include daily curves of financial transaction data and daily patterns of geophysical and environmental data for scalar and vector valued stochastic processes a large number of dependence notions have been proposed mostly involving mixing type distances between algebras in time series analysis measures of dependence based on moments have proven most useful autocovariances and cumulants we introduce a moment based notion of dependence for functional time series which involves m dependence we show that it is applicable to linear as well as nonlinear functional time series then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data we study the estimation of the functional principal components the long run covariance matrix change point detection and the functional linear model we explain when temporal dependence affects the results obtained for i i d functional observations and when these results are robust to weak dependence

functional regression of continuous state distributions abstract in this paper we consider a regression model to study the distributional relationship between economic variables unlike the classical regression dealing exclusively with mean relationship our model can be used to analyze the entire dependent structure in distribution technically we treat density functions as random elements and represent the regression relationship as a compact linear operator in the hilbert spaces of square integrable functions we propose a consistent estimation procedure for our model and develop a test to investigate the dependent structure of moments an empirical example is provided to illustrate how our methodology can be implemented in practical applications

on the p point uncertainty class corresp the p point uncertainty class also known as sakrison s class b is considered although the p point class is not a 2 alternating choquet capacity class it is shown how many of the robust signal processing and communication theory results which have been developed for capacity classes can be applied to the p point class by embedding it in a slightly larger extended p point class

brief paper an analysis of the effects of spectral uncertainty on wiener filtering a representative selection of results from an extensive study of the performance of wiener filtering under spectral uncertainty is presented for a variety of spectral uncertainty models the wiener filter is shown to have undesirable sensitivity to even small deviations from those signal and noise spectral densities which were used to design the filter performance of a robust filter designed to have the best possible performance when the uncertainty is worst is also examined in most cases the robust filter s insensitivity to spectral uncertainty makes it preferable to the traditional wiener filter

nonparametric instrumental regression the focus of this paper is the nonparametric estimation of an instrumental regression function f defined by conditional moment restrictions that stem from a structural econometric model e y f z w 0 and involve endogenous variables y and z and instruments w the function f is the solution of an ill posed inverse problem and we propose an estimation procedure based on tikhonov regularization the paper analyzes identification and overidentification of this model and presents asymptotic properties of the estimated nonparametric instrumental regression function

testing stationarity of functional time series economic and financial data often take the form of a collection of curves observed consecutively over time examples include intraday price curves yield and term structure curves and intraday volatility curves such curves can be viewed as a time series of functions a fundamental issue that must be addressed before an attempt is made to statistically model such data is whether these curves perhaps suitably transformed form a stationary functional time series this paper formalizes the assumption of stationarity in the context of functional time series and proposes several procedures to test the null hypothesis of stationarity the tests are nontrivial extensions of the broadly used tests in the kpss family the properties of the tests under several alternatives including change point and i 1 are studied and new insights present only in the functional setting are uncovered the theory is illustrated by a small simulation study and an application to intraday price curves

intrinsically optimal bayesian robust filtering when designing optimal filters it is often unrealistic to assume that the statistical model is known perfectly the issue is then to design a robust filter that is optimal relative to an uncertainty class of processes robust filter design has been treated from minimax best worst case performance and bayesian best average performance perspectives heretofore the bayesian approach has involved finding a model specific optimal filter one that is optimal for some model in the uncertainty class lifting this constraint we optimize over the full class from which the original optimal filters were obtained for instance over all linear filters by extending the original characteristics determining the filter such as the power spectral density to effective characteristics that apply across the uncertainty class we demonstrate for both linear and morphological filtering that an intrinsically optimal bayesian robust filter can be represented in the same form as the standard solution to the optimal filter except via the effective characteristics solutions for intrinsic bayesian robust filters are more transparent and intuitive than solutions for model specific filters and also less tedious because effective characteristics push through the spectral theory into the bayesian setting whereas solutions in the model specific case depend on grinding out the optimization

minimax robust deconvolution filters under stochastic parametric and noise uncertainties the author consider the design of robust deconvolution filters for linear discrete time systems with stochastic parameter and noise uncertainties it is assumed that some large but bounded uncertainties exist in the driving and measurement noise covariances as well as the second order statistics of stochastic parameters and initial conditions three kinds of minimax sensitivity criteria are used to develop the techniques to the synthesis of minimax deconvolution filters under uncertain linear stochastic systems their approach is based on saddle point theory and the sensitivity analysis of kalman filters the design algorithms give the recursive realization of the minimax deconvolution filters for the time varying uncertain systems under fairly general conditions for the time invariant uncertain case the existence and solutions of steady state deconvolution filters are further developed finally the utility of the minimax design approaches and the properties of the resulting minimax deconvolution filters are illustrated by a numerical example

minimax linear smoothing for capacities abstract minimax linear smoothers are considered for the problem of estimating a homogeneous signal field in an additive orthogonal noise field a minimax game with the quadratic mean estimation error as an objective function is used to formulate this problem uncertainty in signal and noise field spectra is modeled using general nonparametric classes of measures proposed by huber and strassen for the problem of minimax hypothesis testing these classes which are described in terms of choquet alternating capacities of order 2 include the conventional models for spectral uncertainty and admit a general solution to the minimax linear smoothing problem author

joint sampling distribution between actual and estimated classification errors for linear discriminant analysis error estimation must be used to find the accuracy of a designed classifier an issue that is critical in biomarker discovery for disease diagnosis and prognosis in genomics and proteomics this paper presents for what is believed to be the first time the analytical formulation for the joint sampling distribution of the actual and estimated errors of a classification rule the analysis presented here concerns the linear discriminant analysis lda classification rule and the resubstitution and leave one out error estimators under a general parametric gaussian assumption exact results are provided in the univariate case and a simple method is suggested to obtain an accurate approximation in the multivariate case it is also shown how these results can be applied in the computation of condition bounds and the regression of the actual error given the observed error estimate in contrast to asymptotic results the analysis presented here is applicable to finite training data in particular it applies in the small sample settings commonly found in genomics and proteomics applications numerical examples which include parameters estimated from actual microarray data illustrate the analysis throughout

on the mazur ulam problem in linear 2 normed spaces abstract we study the notion of 2 isometry which is suitable to represent the concept of area preserving mapping in linear 2 normed spaces and then prove the mazur ulam problem in linear 2 normed spaces

quantifying the objective cost of uncertainty in complex dynamical systems real world problems often involve complex systems that cannot be perfectly modeled or identified and many engineering applications aim to design operators that can perform reliably in the presence of such uncertainty in this paper we propose a novel bayesian framework for objective based uncertainty quantification uq which quantifies the uncertainty in a given system based on the expected increase of the operational cost that it induces this measure of uncertainty called mocu mean objective cost of uncertainty provides a practical way of quantifying the effect of various types of system uncertainties on the operation of interest furthermore the proposed uq framework provides a general mathematical basis for designing robust operators and it can be applied to diverse applications including robust filtering classification and control we demonstrate the utility and effectiveness of the proposed framework by applying it to the problem of robust structural intervention of gene regulatory networks an important application in translational genomics

on the bias and variability of bootstrap and cross validation estimates of error rate in discrimination problems simulation studies have shown that bootstrap and cross validation estimators of aggregate error rate in discrimination problems have different properties the former having less variability but greater bias we show by use of a local alternative model that the main differences in bias and variability emerge only when the populations are close in this context the bootstrap method has less variability but an order of magnitude greater bias than cross validation