splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

the design of trellis coded mpsk for fading channels set partitioning for optimum code design a previous work on criteria for designing trellis coded mpsk modulation to achieve minimum error probability performance on the rician fading channel see ibid vol 36 no 9 p 1004 1012 sep 1988 is extended it is demonstrated that allowing for multiple symbols per trellis branch i e multiple trellis coded modulation mtcm provides an additional degree of freedom for designing a code to meet the optimization on the fading channel diversities larger than those achievable with conventional trellis codes having the same number of trellis states are now attainable it is under these conditions that mtcm achieves its full potential

fusion of hard and soft information in nonparametric density estimation abstract this paper discusses univariate density estimation in situations when the sample hard information is supplemented by soft information about the random phenomenon these situations arise broadly in operations research and management science where practical and computational reasons severely limit the sample size but problem structure and past experiences could be brought in in particular density estimation is needed for generation of input densities to simulation and stochastic optimization models in analysis of simulation output and when instantiating probability models we adopt a constrained maximum likelihood estimator that incorporates any possibly random soft information through an arbitrary collection of constraints we illustrate the breadth of possibilities by discussing soft information about shape support continuity smoothness slope location of modes symmetry density values neighborhood of known density moments and distribution functions the maximization takes place over spaces of extended real valued semicontinuous functions and therefore allows us to consider essentially any conceivable density as well as convenient exponential transformations the infinite dimensionality of the optimization problem is overcome by approximating splines tailored to these spaces to facilitate the treatment of small samples the construction of these splines is decoupled from the sample we discuss existence and uniqueness of the estimator examine consistency under increasing hard and soft information and give rates of convergence numerical examples illustrate the value of soft information the ability to generate a family of diverse densities and the effect of misspecification of soft information

rates of convergence for bayes and maximum likelihood estimation for mixture of normal densities we study the rates of convergence of the maximum likelihood estimator mle and posterior distribution in density estimation problems where the densities are location or location scale mixtures of normal distributions with the scale parameter lying between two positive numbers the true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub gaussian tails we obtain bounds for hellinger bracketing entropies for this class and from these bounds we deduce the convergence rates of sieve mles in hellinger distance the rate turns out to be log n n where k 1 is a constant that depends on the type of mixtures and the choice of the sieve next we consider a dirichlet mixture of normals as a prior on the unknown density we estimate the prior probability of a certain kullback leibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the hellinger entropy and the concentration rate of the prior the posterior distribution is also seen to converge at the rate log n n in where k now depends on the tail behavior of the base measure of the dirichlet process

on the combinatorics of locally repairable codes via matroid theory this paper provides a link between matroid theory and locally repairable codes lrcs that are either linear or more generally almost affine using this link new results on both lrcs and matroid theory are derived the parameters n k d r delta of lrcs are generalized to matroids and the matroid analog of the generalized singleton bound by gopalan et al for linear lrcs is given for matroids it is shown that the given bound is not tight for certain classes of parameters implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper constructions of classes of matroids with a large span of the parameters n k d r delta and the corresponding local repair sets are given using these matroid constructions new lrcs are constructed with prescribed parameters the existence results on linear lrcs and the nonexistence results on almost affine lrcs given in this paper strengthen the nonexistence and existence results on perfect linear lrcs given by song et al

a semi parametric censored regression estimator abstract this paper introduces a semi parametric method for estimating regression co underlying parent population of errors in censored the method is an example of the method of sieves and it provides simultaneous estimates of the regression coefficients and the density of the underlying parent population in the very simplest terms the underlying unknown density is approximated by a spline with mesh size approaching zero with the sample size the values of the density at the knots are then added to the list of the usual unknown parameters in a censored regression model e g the regression coefficients and scale parameter a quasi likelihood function using the approximate spline density is then maximized over all the parameters mentioned above the method is shown to result in strongly consistent parameter estimates

asymptotic optimality of generalized cl cross validation and generalized cross validation in regression with heteroskedastic errors abstract the problem considered here is that of using a data driven procedure to select a good estimate from a class of linear estimates indexed by a discrete parameter in contrast to other papers on this subject we consider models with heteroskedastic errors the results apply to model selection problems in linear regression and to nonparametric regression estimation via series estimators nearest neighbor estimators and local regression estimators among others generalized c l gc l cross validation cv and generalized cross validation gcv procedures are analyzed the gc l and cv criteria are shown to be asymptotically optimal under general conditions a feasible version of gc l however is available only in some applications the gcv criterion is found to be asymptotically optimal only under a condition that is satisfied in some applications but not in others for example it is satisfied in the nearest neighbor estimation context but not in the series estimation local regression estimation or model selection contexts thus the cv criterion is the only feasible criterion of the three that is asymptotically optimal under general conditions the proofs rely heavily on results of li 1987

on bayesian consistency we consider a sequence of posterior distributions based on a data dependent prior which we shall refer to as a pseudoposterior distribution and establish simple conditions under which the sequence is hellinger consistent it is shown how investigations into these pseudo posteriors assist with the understanding of some true posterior distributions including polya trees the infinite dimensional exponential family and mixture models

canonical analysis of two euclidean subspaces and its applications abstract we develop a canonical analysis both without constraints and under constraints for subspaces of euclidean space we also look into a canonical analysis of operators we then apply the definitions and some of the results to the field of probability and statistics

penalized weighted least squares image reconstruction for positron emission tomography presents an image reconstruction method for positron emission tomography pet based on a penalized weighted least squares pwls objective for pet measurements that are precorrected for accidental coincidences the author argues statistically that a least squares objective function is as appropriate if not more so than the popular poisson likelihood objective the author proposes a simple data based method for determining the weights that accounts for attenuation and detector efficiency a nonnegative successive over relaxation sor algorithm converges rapidly to the global minimum of the pwls objective quantitative simulation results demonstrate that the bias variance tradeoff of the pwls sor method is comparable to the maximum likelihood expectation maximization ml em method but with fewer iterations and is improved relative to the conventional filtered backprojection fbp method qualitative results suggest that the streak artifacts common to the fbp method are nearly eliminated by the pwls sor method and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over fbp

repairing reed solomon codes we study the performance of reed solomon rs codes for the exact repair problem in distributed storage our main result is that in some parameter regimes reed solomon codes are optimal regenerating codes among maximum distance separable mds codes with linear repair schemes moreover we give a characterization of mds codes with linear repair schemes which holds in any parameter regime and which can be used to give non trivial repair schemes for rs codes in other settings more precisely we show that for k dimensional rs codes whose evaluation points are a finite field of size n there are exact repair schemes with bandwidth n 1 log n 1 n k bits and that this is optimal for any mds code with a linear repair scheme in contrast the naive commonly implemented repair algorithm for this rs code has bandwidth k log n bits when the entire field is used as evaluation points the number of nodes n is much larger than the number of bits per node which is o log n and so this result holds only when the degree of sub packetization is small however our method applies in any parameter regime and to illustrate this for high levels of sub packetization we give an improved repair scheme for a specific 14 10 rs code used in the facebook hadoop analytics cluster

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

rates of convergence for the estimates of the optimal transformations of variables we consider here spline estimates of the optimal transformations of variables for multiple correlation and regression as dealt with in a recent paper by breiman and friedman we show that we can construct estimates of the optimal transformations which have the same optimal rate of convergence as in the usual nonparametric estimation of a univariate function

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

on measuring internal dependence in a set of random variables on considere un analogue multivariable de la correlation maximale qui consiste a transformer chacune des variables de telle sorte que les plus grandes sommes partielles de valeurs propres de la matrice de correlation resultante soient maximisees

performance of rs coded m ary modulation with and without symbol overlapping in this paper we present analytical bit error probability results for m ary modulation concatenated with reed solomon rs codes the analysis of bit error probability is nontrivial as the number of bits per symbol for the rs codes may not be an integer multiple of the number of bits per symbol for a modulation symbol we propose a markov chain technique which allows analytical evaluation of the bit error probability for such cases the performance of rs coding with coherent biorthogonal coherent non coherent orthogonal modulation over an additive white gaussian noise awgn channel is evaluated simulation of the bit error probability of rs code concatenated with a nordstrom robinson nr code as an inner code is performed and compared with the case of biorthogonal modulation from the results we notice that a stronger inner code gives better bit error probability in addition the throughput of the coded system with biorthogonal modulation over an awgn channel is discussed for a rayleigh flat fading and block fading channel we analyze the bit error probability of rs codes concatenated with biorthogonal modulation from the result we notice that a stronger outer code gives a better bit error probability for the case of rayleigh flat fading channel

convergence and consistency results for self modeling nonlinear regression on propose une methode d estimation simultanee et un algorithme iteratif simplifiant le calcul des estimateurs

asymptotic normality of series estimators for nonparametric and semiparametric regression models this paper establishes the asymptotic normality of series estimators for nonparametric regression models gallant s fourier flexible form estimators trigonometric series estimators and polynomial series estimators are prime examples of the estimators covered by the results the results apply to a wide variety of estimates in the regression model under consideration including derivatives and integrals of the regression function the errors in the model may be homoskedastic or heteroskedastic the paper also considers series estimators for additive interactive regression semiparametric regression and semiparametric index regression models and shows them to be consistent and asymptotically normal copyright 1991 by the econometric society

efficient estimation of average treatment effects using the estimated propensity score we are interested in estimating the average effect of a binary treatment on a scalar outcome if assignment to the treatment is exogenous or unconfounded that is independent of the potential outcomes given covariates biases associated with simple treatmentcontrol average comparisons can be removed by adjusting for differences in the covariates rosenbaum and rubin 1983 show that adjusting solely for differences between treated and control units in the propensity score removes all biases associated with differences in covariates although adjusting for differences in the propensity score removes all the bias this can come at the expense of efficiency as shown by hahn 1998 heckman ichimura and todd 1998 and robins mark and newey 1992 we show that weighting by the inverse of a nonparametric estimate of the propensity score rather than the true propensity score leads to an efficient estimate of the average treatment effect we provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

resolution based complexity control for gaussian mixture models in the domain of unsupervised learning mixtures of gaussians have become a popular tool for statistical modeling for this class of generative models we present a complexity control scheme which provides an effective means for avoiding the problem of overfitting usually encountered with unconstrained mixtures of gaussians in high dimensions according to some prespecified level of resolution as implied by a fixed variance noise model the scheme provides an automatic selection of the dimensionalities of some local signal subspaces by maximum likelihood estimation together with a resolution based control scheme for adjusting the number of mixture components we arrive at an incremental model refinement procedure within a common deterministic annealing framework which enables an efficient exploration of the model space the advantages of the resolution based framework are illustrated by experimental results on synthetic and high dimensional real world data

iterated residuals and time varying covariate effects in cox regression summary the cox proportional hazards model which is widely used for the analysis of treatment and prognostic effects with censored survival data makes the assumption that the hazard ratio is constant over time nonparametric estimators have been developed for an extended model in which the hazard ratio is allowed to change over time estimators based on residuals are appealing as they are easy to use and relate in a simple way to the more restricted cox model estimator after fitting a cox model and calculating the residuals one can obtain a crude estimate of the time varying coefficients by adding a smooth of the residuals to the initial constant estimate treating the crude estimate as the fit one can re estimate the residuals iteration leads to consistent estimation of the nonparametric time varying coefficients this approach leads to clear guidelines for residual analysis in applications the results are illustrated by an analysis of the medical research council s myeloma trials and by simulation

the reduced parzen classifier the parzen density estimate is known to be an effective tool for estimating the bayes error given a set of training samples from the class distributions an algorithm is developed to select a given number of representative samples whose parzen density estimate closely matches that of the entire sample set using this reduced representative set a piecewise quadratic classifier which provides nearly optimal performance is designed

estimation of the binary response model using a mixture of distributions estimator mod abstract in this paper we develop a semiparametric sieve estimator which is termed a mixture of distributions estimator mod to estimate a binary response model when the distribution of the errors is unknown the estimator of the distribution function is composed of a mixture of smooth distributions where the number of mixtures increases with the sample size the model is semiparametric because it is assumed that a parametric index type restriction holds optimal rates of convergence are established for the distribution function under the l 2 norm and conditions are derived under which estimates of the parametric component are asymptotically normal an appealing feature about mod is that it is possible to restrict the estimator of the distribution function a priori to be smooth nonnegative nondecreasing and to integrate to one this has important practical and theoretical implications

data based optimal smoothing of orthogonal series density estimates let f be a density possessing some smoothness properties and let x 1 cdots x n be independent observations from f some desirable properties of orthogonal series density estimates f n m lambda of f of the form f n m lambda t sum n nu 1 frac hat f nu 1 lambda nu 2m phi nu t where phi nu is an orthonormal sequence and hat f nu 1 n sum n j 1 phi nu x j is an estimate of f nu int phi nu t f t dt are discussed the parameter lambda plays the role of a bandwidth or smoothing parameter and m controls a shape factor the major novel result of this note is a simple method for estimating lambda and m from the data in an objective manner to minimize integrated mean square error the results extend to multivariate estimates

semiparametric estimation of normal mixture densities a semiparametric method for estimating densities of normal mean mixtures is presented this consistent data driven method of estimation is based on probability spacings the estimation technique involves iteratively fixing the standard deviation of the normal kernel that serves as a smoothing parameter and then maximizing a function of the probability spacings over all mixing distributions based on the distribution of uniform spacings a distribution free goodness of fit criterion is developed to guide the selection of the smoothing parameter the result is a set of consistent estimators indexed by a range of smoothing parameters empirical process results are used to prove consistency

alphabet dependent bounds for linear locally repairable codes based on residual codes locally repairable codes lrcs have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others several works have thus been carried out to understand the optimal rate distance tradeoff but only recently the size of the alphabet has been taken into account in this paper a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield mds codes then a new alphabet dependent bound is derived which applies both to the new definition and the initial definition of locality the new bound is based on consecutive residual codes and intrinsically uses the griesmer bound a special case of the bound yields both the extension of the cadambe mazumdar bound and the singleton type bound for codes with locality r delta implying that the new bound is at least as good as these bounds furthermore an upper bound on the asymptotic rate distance tradeoff of lrcs is derived and yields the tightest known upper bound for large relative minimum distances achievability results are also provided by deriving the locality of the family of simplex codes together with a few examples of optimal codes

bandwidth efficient concatenated coding schemes for fading channels concatenated codes consisting of trellis inner codes and reed solomon outer codes are considered to achieve large coding gains with small bandwidth expansion in the presence of frequency nonselective slow rician fading both errors only and errors erasures decoding algorithms for outer codes are applied new upper bounds on bit error probability performance in the presence of fading are obtained and compared with simulation results for zero channel memory the effect of interleaving in eliminating channel memory is investigated the performance gains that are achieved by the coding scheme relative to the reference uncoded systems are illustrated via some examples

randomized mixture models for probability density approximation and estimation abstract neural networks nns with random weights are an interesting alternative to conventional nns that are used more for data modeling the random vector functional link rvfl network is an established and theoretically well grounded randomized learner a key theoretical result for rvfl networks is that they provide universal approximation for continuous maps in expectation with respect to the square integral norm we specialize and modify this result and show that rfvl networks can provide functional approximations that converge in kullback leibler divergence when the target function is a probability density function expanding on the approximation results we demonstrate the rfvl networks lead to a simple randomized mixture model mm construction for density estimation from sample data an expectation maximization em algorithm is derived for the maximum likelihood estimation of our randomized mm the em algorithm is proved to be globally convergent and the maximum likelihood estimator is proved to be consistent a set of simulation studies is given to provide empirical evidence towards our approximation and density estimation results

additivity tests for nonlinear autoregression summary additivity is commonly used in the statistical literature to simplify data analysis especially in analysis of variance and in multivariate smoothing in this paper we propose three procedures for testing additivity in nonlinear time series analysis the first procedure combines some smoothing techniques with analysis of variance the second is a lagrange multiplier test using nonparametric estimation and the third is a permutation test which uses smoothing techniques to obtain the test statistic and its reference distribution we investigate properties of the proposed tests and use simulation to check their performance in finite samples applications of the tests to nonlinear time series analysis are discussed and illustrated by real examples

multiple symbol differentially detected multilevel codes for the rayleigh fading channel conventional differential detection is known to perform poorly on the fast fading rayleigh fading channel due to the rapid variation of the channel state the technique of multiple symbol differential detection msdd improves the error performance by detecting over a sequence of symbols and exploiting the inherent correlation of the fading process to further reduce the error rate we need to introduce some form of coding successful coding for the rayleigh fading channel requires interleaving to decorrelate the channel a conflicting requirement to msdd we propose a solution consisting of interleaving blocks of l m ary phase shift keying mpsk symbols that are multiple symbol differentially detected the blocks are treated as elements of an m sup l symbol alphabet over which a multilevel code based on geometrically uniform partitions is defined

a fuzzy nonparametric segmentation framework for dti and mri analysis with applications to dti tract extraction this paper presents a novel fuzzy segmentation method for diffusion tensor dt and magnetic resonance mr images typical fuzzy segmentation schemes e g those based on fuzzy c means fcm incorporate gaussian class models that are inherently biased towards ellipsoidal clusters characterized by a mean element and a covariance matrix tensors in fiber bundles however inherently lie on specific manifolds in riemannian spaces unlike fcm based schemes the proposed method represents these manifolds using nonparametric data driven statistical models the paper describes a statistically sound consistent technique for nonparametric modeling in riemannian dt spaces the proposed method produces an optimal fuzzy segmentation by maximizing a novel information theoretic energy in a markov random field framework results on synthetic and real dt and mr images show that the proposed method provides information about the uncertainties in the segmentation decisions which stem from imaging artifacts including noise partial voluming and inhomogeneity by enhancing the nonparametric model to capture the spatial continuity and structure of the fiber bundle we exploit the framework to extract the cingulum fiber bundle typical tractography methods for tract delineation incorporating thresholds on fractional anisotropy and fiber curvature to terminate tracking can face serious problems arising from partial voluming and noise for these reasons tractography often fails to extract thin tracts with sharp changes in orientation such as the cingulum the results demonstrate that the proposed method extracts this structure significantly more accurately as compared to tractography

optimal locally repairable codes and connections to matroid theory petabyte scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency classical codes such as reed solomon rs are highly sub optimal for distributed environments due to their high overhead during single failure events locally repairable codes lrcs form a new family of codes that are repair efficient in particular lrcs minimize the number of nodes participating in single node repairs fundamental bounds and methods for explicitly constructing lrcs suitable for deployment in distributed storage clusters are not fully understood and currently form an active area of research in this paper we present an explicit lrc that is simple to construct and is optimal for a specific set of coding parameters our construction is based on grouping rs symbols and then adding extra simple parities that allow for small repair locality for the analysis of the optimality of the code we derive a new result on the matroid represented by the code s generator matrix

sinh arcsinh distributions we introduce the sinh arcsinh transformation and hence by applying it to a generating distribution with no parameters other than location and scale usually the normal a new family of sinh arcsinh distributions this four parameter family has symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution the central place of the normal distribution in this family affords likelihood ratio tests of normality that are superior to the state of the art in normality testing because of the range of alternatives against which they are very powerful likelihood ratio tests of symmetry are also available and are very successful three parameter symmetric and asymmetric subfamilies of the full family are also of interest heavy tailed symmetric sinh arcsinh distributions behave like johnson su distributions while their light tailed counterparts behave like sinh normal distributions the sinh arcsinh family allowing a seamless transition between the two via the normal controlled by a single parameter the sinh arcsinh family is very tractable and many properties are explored likelihood inference is pursued including an attractive reparameterization illustrative examples are given a multivariate version is considered options and extensions are discussed

nonparametric estimation via empirical risk minimization a general notion of universal consistency of nonparametric estimators is introduced that applies to regression estimation conditional median estimation curve fitting pattern recognition and learning concepts general methods for proving consistency of estimators based on minimizing the empirical error are shown in particular distribution free almost sure consistency of neural network estimates and generalized linear estimators is established

bounds on the parameters of locally recoverable codes a locally recoverable code lrc code is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set in this paper we derive new finite length and asymptotic bounds on the parameters of lrc codes for lrc codes with a single recovering set for every coordinate we derive an asymptotic gilbert varshamov type bound for lrc codes and find the maximum attainable relative distance of asymptotically good lrc codes similar results are established for lrc codes with two disjoint recovering sets for every coordinate for the case of multiple recovering sets the availability problem we derive a lower bound on the parameters using expander graph arguments finally we also derive finite length upper bounds on the rate and the distance of lrc codes with multiple recovering sets

bounds on the size of locally recoverable codes in a locally recoverable or repairable code any symbol of a codeword can be recovered by reading only a small constant number of other symbols the notion of local recoverability is important in the area of distributed storage where a most frequent error event is a single storage node failure erasure a common objective is to repair the node by downloading data from as few other storage nodes as possible in this paper we bound the minimum distance of a code in terms of its length size and locality unlike the previous bounds our bound follows from a significantly simple analysis and depends on the size of the alphabet being used it turns out that the binary simplex codes satisfy our bound with equality hence the simplex codes are the first example of an optimal binary locally repairable code family we also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds

on the locality of codeword symbols consider a linear n k d q code c we say that the ith coordinate of c has locality r if the value at this coordinate can be recovered from accessing some other r coordinates of c data storage applications require codes with small redundancy low locality for information coordinates large distance and low locality for parity coordinates in this paper we carry out an in depth study of the relations between these parameters we establish a tight bound for the redundancy n k in terms of the message length the distance and the locality of information coordinates we refer to codes attaining the bound as optimal we prove some structure theorems about optimal codes which are particularly strong for small distances this gives a fairly complete picture of the tradeoffs between codewords length worst case distance and locality of information symbols we then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes using our structure theorem we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings we prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance

a smoothed em approach to indirect estimation problems with particular reference to stereology and emission tomography there are many practical problems where the observed data are not drawn directly from the density g of real interest but rather from another distribution derived from g by the application of an integral operator the estimation of g then entails both statistical and numerical difficulties a natural statistical approach is by maximum likelihood conveniently implemented using the em algorithm but this provides unsatisfactory reconstructions of g in this paper we modify the maximum likelihood em approach by introducing a simple smoothing step at each em iteration in our experience this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration some theoretical background is given that relates this smoothed em algorithm to a maximum penalized likelihood approach two applications are considered in detail the first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium the second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography for this problem we also suggest several technical improvements on existing methodology

multivariate analysis with linearizable regressions we study the class of multivariate distributions in which all bivariate regressions can be linearized by separate transformation of each of the variables this class seems more realistic than the multivariate normal or the elliptical distributions and at the same time its study allows us to combine the results from multivariate analysis with optimal scaling and classical multivariate analysis in particular a two stage procedure which first scales the variables optimally and then fits a simultaneous equations model is studied in detail and is shown to have some desirable properties

distribution free estimation of heteroskedastic binary response models using probit logit criterion functions abstract in this paper estimators for distribution free heteroskedastic binary response models are proposed the estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models such as probit logit exhibiting multiplicative heteroskedasticity the first proposed estimator is based on the observational equivalence between the two models and is a semiparametric sieve estimator see e g gallant and nychka 1987 ai and chen 2003 and chen et al 2005 for the regression coefficients based on maximizing standard logit probit criterion functions such as nlls and mle this procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously the second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model manski 1975 manski 1985 horowitz 1992 and the standard parametric probit logit nlls estimator this estimator has the advantage of being implementable with standard software packages such as stata distribution theory is developed for both estimators and a monte carlo study indicates they both perform well in finite samples

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

adaptive mixture density estimation abstract a recursive nonparametric method is developed for performing density estimation derived from mixture models kernel estimation and stochastic approximation the asymptotic performance of the method dubbed adaptive mixtures priebe and marchette pattern recognition24 1197 1209 1991 for its data driven development of a mixture model approximation to the true density is investigated using the method of sieves simulations are included indicating convergence properties for some simple examples

nonlinear additive arx models abstract we consider in this article a class of nonlinear additive autoregressive models with exogenous variables for nonlinear time series analysis and propose two modeling procedures for building such models the procedures proposed use two backfitting techniques the ace and bruto algorithms to identify the nonlinear functions involved and use the methods of best subset regression and variable selection in regression analysis to determine the final model simulated and real examples are used to illustrate the analysis

practical bayesian density estimation using mixtures of normals abstract mixtures of normals provide a flexible model for estimating densities in a bayesian framework there are some difficulties with this model however first standard reference priors yield improper posteriors second the posterior for the number of components in the mixture is not well defined if the reference prior is used third posterior simulation does not provide a direct estimate of the posterior for the number of components we present some practical methods for coping with these problems finally we give some results on the consistency of the method when the maximum number of components is allowed to grow with the sample size

constructions and properties of linear locally repairable codes in this paper locally repairable codes with all symbol locality are studied methods to modify already existing codes are presented it is also shown that with high probability a random matrix with a few extra columns guaranteeing the locality property is a generator matrix for a locally repairable code with a good minimum distance the proof of the result provides a constructive method to find locally repairable codes finally constructions of three infinite classes of optimal vector linear locally repairable codes over a small alphabet independent of the code size are given

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

on heteroscedastic hazards regression models theory and application summary a class of non proportional hazards regression models is considered to have hazard specifications consisting of a power form of cross effects on the base line hazard function the primary goal of these models is to deal with settings in which heterogeneous distribution shapes of survival times may be present in populations characterized by some observable covariates although effects of such heterogeneity can be explicitly seen through crossing cumulative hazards phenomena in k sample problems they are barely visible in a one sample regression setting hence heterogeneity of this kind may not be noticed and more importantly may result in severely misleading inference this is because the partial likelihood approach cannot eliminate the unknown cumulative base line hazard functions in this setting for coherent statistical inferences a system of martingale processes is taken as a basis with which together with the method of sieves an overidentified estimating equation approach is proposed a pearson x2 type of goodness of fit testing statistic is derived as a by product an example with data on gastric cancer patients survival times is analysed

nonparametric maximum likelihood estimation by the method of sieves maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space for example the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an operatorname iid sample the maximum of the likelihood is not attained by any density in this example as in many other examples the parameter space positive functions with area one is too big but the likelihood method can often be salvaged if we first maximize over a constrained subspace of the parameter space and then relax the constraint as the sample size grows this is grenander s method of sieves application of the method sometimes leads to new estimators for familiar problems or to a new motivation for an already well studied technique we will establish some general consistency results for the method and then we will focus on three applications

adaptive model selection using empirical complexities given n independent replicates of a jointly distributed pair x y rd r we wish to select from a fixed sequence of model classes f1 f2 a deterministic prediction rule f rd r whose risk is small we investigate the possibility of empirically assessing the complexity of each model class that is the actual difficulty of the estimation problem within each class the estimated complexities are in turn used to define an adaptive model selection procedure which is based on complexity penalized empirical risk the available data are divided into two parts the first is used to form an empirical cover of each model class and the second is used to select a candidate rule from each cover based on empirical risk the covering radii are determined empirically to optimize a tight upper bound on the estimation error an estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk a distinguishing feature of the approach is that the complexity of each model class is assessed empirically based on the size of its empirical cover finite sample performance bounds are established for the estimates and these bounds are applied to several non parametric estimation problems the estimates are shown to achieve a favorable tradeoff between approximation and estimation error and to perform as well as if the distribution dependent complexities of the model classes were known beforehand in addition it is shown that the estimate can be consistent and even possess near optimal rates of convergence when each model class has an infinite vc or pseudo dimension for regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence appears in annals of statistics vol 27 1830 1864 ams 1991 subject classifications primary 62g07 62g20 secondary 62h30

rates of convergence for the gaussian mixture sieve gaussian mixtures provide a convenient method of densityestimation that lies somewhere between parametric models and kernel densityestimators when the number of components of the mixture is allowed to increase as sample size increases the model is called a mixture sieve we establish a bound on the rate of convergence in hellinger distance for densityestimation using the gaussian mixture sieve assuming that the true density is itself a mixture of gaussians the underlying mixing measure of the true densityis not necessarilyassumed to have finite support computing the rate involves some delicate calculations since the size of the sieve as measured bybracketing entropy and the saturation rate cannot be found using standard methods when the mixing measure has compact support using kn n 2 3 log n 1 3 components in the mixture yields a rate of order log n 1 6 n1 6 for every 0 the rates depend heavilyon the tail behavior of the true density the sensitivity to the tail behavior is diminished byusing a robust sieve which includes a long tailed component in the

convergence rates for density estimation with bernstein polynomials mixture models for density estimation provide a very useful set up for the bayesian or the maximum likelihood approach for a density on the unit interval mixtures of beta densities form a flexible model the class of bernstein densities is a much smaller subclass of the beta mixtures defined by bernstein polynomials which can approximate any continuous density a bernstein polynomial prior is obtained by putting a prior distribution on the class of bernstein densities the posterior distribution of a bernstein polynomial prior is consistent under very general conditions in this article we present some results on the rate of convergence of the posterior distribution if the underlying distribution generating the data is itself a bernstein density then we show that the posterior distribution converges at nearly parametric rate log n sqrt n for the hellinger distance if the true density is not of the bernstein type we show that the posterior converges at a rate n 1 3 log n 5 6 provided that the true density is twice differentiable and bounded away from 0 similar rates are also obtained for sieve maximum likelihood estimates these rates are inferior to the pointwise convergence rate of a kernel type estimator we show that the bayesian bootstrap method gives a proxy for the posterior distribution and has a convergence rate at par with that of the kernel estimator

an efficient uniform cost algorithm applied to distance transforms the uniform cost algorithm is a special case of the a algorithm for finding the shortest paths in graphs in the uniform cost algorithm nodes are expanded in order of increasing cost an efficient version of this algorithm is developed for integer cost values nodes are sorted by storing them at predefined places bucket sort keeping the overhead low the algorithm is applied to general distance transformation a constrained distance transform is an operation which calculates at each pixel of an image the distance to the nearest pixel of a reference set distance being defined as minimum path length the uniform cost algorithm in the constrained case proves to be the best solution for distance transformation it is fast the processing time is independent of the complexity of the image and memory requirements are moderate

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

density estimation with confidence sets exemplified by superclusters and voids in the galaxies abstract a method is presented for forming both a point estimate and a confidence set of semiparametric densities the final product is a three dimensional figure that displays a selection of density estimates for a plausible range of smoothing parameters the boundaries of the smoothing parameter are determined by a nonparametric goodness of fit test that is based on the sample spacings for each value of the smoothing parameter our estimator is selected by choosing the normal mixture that maximizes a function of the sample spacings a point estimate is selected from this confidence set by using the method of cross validation an algorithm to find the mixing distribution that maximizes the spacings functional is presented these methods are illustrated with a data set from the astronomy literature the measurements are velocities at which galaxies in the corona borealis region are moving away from our galaxy if the galaxies are clustered the velocity density will be multimodal with clusters correspond

efficient estimation of semiparametric multivariate copula models we propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions a joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions this class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable and its circumvention of the curse of dimensionality associated with purely nonparametric multivariate distributions we show that the plug in sieve maximum likelihood estimators mles of all smooth functionals including the finite dimensional copula parameters and the unknown marginal distributions are semiparametrically efficient and that their asymptotic variances can be estimated consistently moreover prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve fu

the dimensionality reduction principle for generalized additive models soit x y une paire de variables aleatoires telles que x x 1 x j s etende sur c 0 1 j la distribution conditionnelle de y etant donne x x est supposee appartenir a une famille exponentielle appropriee ayant pour parametre r f x denote la dependance de vis a vis de x et f l approximation additive de f ayant la log vraisemblance esperee maximum possible

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free

some asymptotic results for learning in single hidden layer feedforward network models abstract we investigate the properties of a recursive estimation procedure the method of back propagation for a class of nonlinear regression models single hidden layer feedforward network models recently developed by cognitive scientists the results follow from more general results for a class of recursive m estimators obtained using theorems of ljung 1977 and walk 1977 for the method of stochastic approximation conditions are given ensuring that the back propagation estimator converges almost surely to a parameter value that locally minimizes expected squared error loss provided the estimator does not diverge and that the back propagation estimator is asymptotically normal when centered at this minimizer this estimator is shown to be statistically inefficient and a two step procedure that has efficiency equivalent to that of nonlinear least squares is proposed practical issues are illustrated by a numerical example involving approximation of the henon map