distribution estimation consistent in total variation and in two types of information divergence the problem of the nonparametric estimation of a probability distribution is considered from three viewpoints the consistency in total variation the consistency in information divergence and consistency in reversed order information divergence these types of consistencies are relatively strong criteria of convergence and a probability distribution cannot be consistently estimated in either type of convergence without any restrictions on the class of probability distributions allowed histogram based estimators of distribution are presented which under certain conditions converge in total variation in information divergence and in reversed order information divergence to the unknown probability distribution some a priori information about the true probability distribution is assumed in each case as the concept of consistency in information divergence is stronger than that of convergence in total variation additional assumptions are imposed in the cases of informational divergences

kernel density estimation for linear processes in this paper we provide a detailed characterization of the asymptotic behavior of kernel density estimators for one sided linear processes the conjecture that asymptotic normality for the kernel density estimator holds under short range dependence is proved under minimal assumptions on bandwidths we also depict the dichotomous and trichotomous phenomena for various choices of bandwidths when the process is long range dependent

nearest neighbor classification with dependent training sequences the asymptotic classification risk for nearest neighbor procedures is well understood in the case of i i d training sequences in this article we generalize these results to a class of dependent models including hidden markov models in the case where the observed patterns have lebesgue densities the asymptotic risk takes the same expression as in the i i d case for discrete distributions we show that the asymptotic risk depends on the rule used for breaking ties of equal distances

density estimation under long range dependence dehling and taqqu established the weak convergence of the empirical process for a long range dependent stationary sequence under gaussian subordination we show that the corresponding density process based on kernel estimators of the marginal density converges weakly with the same normalization to the derivative of their limiting process the phenomenon which carries on for higher derivatives and for functional laws of the iterated logarithm is in contrast with independent or weakly dependent situations where the density process cannot be tight in the usual function spaces with supremum distances

on the asymptotic expansion of the empirical process of long memory moving averages let x n i 1 a i e n i where the e i are iid with mean 0 and finite fourth moment and the a i are regularly varying with index where 1 2 1 so that x n has long range dependence this covers an important class of the fractional arima process for r 0 let y n r n n 1 1 j1 jr r s 1 a js e n js y n 0 n 2 n r var y n r and f r the rth derivative of the distribution function of x n the y n r are uncorrelated and are stochastically decreasing in r for any positive integer p 2 1 1 it is shown under mild regularity conditions that with probability 1 n n 1 i x n x p r 0 1 r f r x y n r o n n p uniformly for all x r o 1 2 1 2 p 1 2 this generalizes a host of existing results and provides the vehicle for a number of statistical applications

statistical properties of chaotic systems the theory of smooth dynamical systems and the theory of abstract dynamical systems ergodic theory although having the same roots have for many years developed quite independently of one another these theories have now matured to the point where they can be combined to shed light on the nature of chaotic behavior

on estimating a density using hellinger distance and some other strange facts on s interesse ici aux possibles vitesses d estimation d une densite a support compact dans m sous des hypotheses de regularite lorsque la perte est mesuree par le carre de la distance de hellinger on regardera aussi le cas connu des normes mathbb l q pour 1 q 2 et le risque est le risque minimax sur la famille on donne une methode generale permettant de traiter les problemes dans le cadre de la theorie de l approximation sous des conditions concernant l entropie metrique et l e capacite des familles a estimer les rapports entre regularite et entropie metrique etant bien connus nous pourrons aussi traiter les cas classiques et d autres qui le sont moins sous des conditions de bornes inferieures les vitesses sont celles observees pour la norme mathbb l q mais elles different dans le cas general on montre aussi que les restrictions sur la compacite du support ou la regularite sont indispensables et que leur absence mene a l impossibilite d obtenir une estimation raisonnable en ce sens que n importe quelle suite d estimateurs sera arbitrairement mauvaise en un point au moins un resultat analogue est vrai sous des conditions de regularite

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation

recursive probability density estimation for weakly dependent stationary processes recursive estimation of the univariate probability density function f x for stationary processes x j is considered quadratic mean convergence and asymptotic normality for density estimators f n x are established for strong mixing and for asymptotically uncorrelated processes x j recent results for nonrecursive density estimators are extended to the recursive case

optimization of barron density estimates we investigate a nonparametric estimator of the probability density introduced by barron 1988 1989 earlier papers established its consistency in a strong sense e g in the expected information divergence or expected chi square divergence this paper pays main attention to the expected chi square divergence criterion we give a new motivation of the barron estimator by showing that a maximum likelihood estimator mle of a density from a family important in practice is consistent in expected information divergence but not in expected chi square divergence we also present new and practically applicable conditions of consistency in the expected chi square divergence main attention is paid to optimization in the sense of the mentioned criterion of the two objects specifying the barron estimator the dominating probability density and the decomposition of the observation space into finitely many bins both problems are explicitly solved under certain regularity assumptions about the estimated density a simulation study illustrates the results in exponential rayleigh and weibull families

density estimation and random variate generation using multilayer networks in this paper we consider two important topics density estimation and random variate generation we present a framework that is easily implemented using the familiar multilayer neural network first we develop two new methods for density estimation a stochastic method and a related deterministic method both methods are based on approximating the distribution function the density being obtained by differentiation in the second part of the paper we develop new random number generation methods our methods do not suffer from some of the restrictions of existing methods in that they can be used to generate numbers from any density provided that certain smoothness conditions are satisfied one of our methods is based on an observed inverse relationship between the density estimation process and random number generation we present two variants of this method a stochastic and a deterministic version we propose a second method that is based on a novel control formulation of the problem where a controller network is trained to shape a given density into the desired density we justify the use of all the methods that we propose by providing theoretical convergence results in particular we prove that the l sub spl infin convergence to the true density for both the density estimation and random variate generation techniques occurs at a rate o log log n n sup 1 spl epsi 2 where n is the number of data points and spl epsi can be made arbitrarily small for sufficiently smooth target densities this bound is very close to the optimally achievable convergence rate under similar smoothness conditions also for comparison the l sub 2 root mean square rms convergence rate of a positive kernel density estimator is o n sup 2 5 when the optimal kernel width is used we present numerical simulations to illustrate the performance of the proposed density estimation and random variate generation methods in addition we present an extended introduction and bibliography that serves as an overview and reference for the practitioner

efficiency of a kernel density estimator under an autoregressive dependence model abstract the problem of estimating the probability density function of a strictly stationary process is considered to study the effect of a dependence structure on the efficiency of a kernel density estimator the mean integrated squared error mise of the fourier integral estimator fie is derived on the assumption that the observed data are generated by a first order autoregressive process numerical results for the normal and cauchy densities show that even moderate departures from independence can lead to a considerable loss in efficiency of the fie in addition to efficiency considerations the issue of determining an optimal smoothing parameter for the fie under the autoregressive model is addressed

no empirical probability measure can converge in the total variation sense for all distributions for any sequence of empirical probability measures n on the borel sets of the real line and any 0 there exists a singular continuous probability measure such that inf sup n 9m n a a a 1 2 almost surely

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

order estimation of markov chains estimators spl chi sub n x sub 0 x sub 1 x sub n are described which when applied to an unknown stationary process taking values from a countable alphabet spl chi converge almost surely to k in case the process is a kth order markov chain and to infinity otherwise

multivariate analysis by data depth descriptive statistics graphics and inference with discussion and a rejoinder by liu and singh a data depth can be used to measure the depth or outlyingness of a given multivariate sample with respect to its underlying distribution this leads to a natural center outward ordering of the sample points based on this ordering quantitative and graphical methods are introduced for analyzing multivariate distributional characteristics such as location scale bias skewness and kurtosis as well as for comparing inference methods all graphs are one dimensional curves in the plane and can be easily visualized and interpreted a sunburst plot is presented as a bivariate generalization of the box plot dd depth versus depth plots are proposed and examined as graphical inference tools some new diagnostic tools for checking multivariate normality are introduced one of them monitors the exact rate of growth of the maximum deviation from the mean while the others examine the ratio of the overall dispersion to the dispersion of a certain central region the affine invariance property of a data depth also leads to appropriate invariance properties for the proposed statistics and methods

singular perturbation modelling of continuous and discrete physical systems the major shortcoming of the singular perturbation theory preventing its application to practical problems is that the system described by state equations is seldom given in the singularly perturbed form this paper presents a systematic procedure which converts the state equations of a large scale time invariant system to singularly perturbed form assuming that such a classification is possible this is accomplished by appropriate scaling and grouping of the state variables into slow and fast groups while retaining as many physical state variables as possible in the new state description an improved slow fast grouping of the states can result by allowing linear combination of certain available physical variables as new state variables several examples are given to illustrate the method

singular perturbation modelling of large scale systems with multi time scale property the problem is investigated of singular perturbation modelling which plays an important role in cases of the application of the singular perturbation method to the design and analysis of large scale systems in both continuous and discrete time versions modelling concepts based on structural properties are studied and computer oriented algorithms are developed the results of two time scale systems are extended to multi time scale cases furthermore in order to verify the validity of these studies some numerical examples are given

recursive density estimation under dependence recursive estimators of the density of weakly dependent random variables are studied under certain absolute regularity and strong mixing conditions uniform strong consistency of the density estimators is established and their rates of convergence are obtained this study is concerned more with the almost sure uniform consistency of a sequence and its rate of convergence than with pointwise convergence since parameter estimation in time series analysis is often carried out under the gaussian assumption it is useful to check whether or not the density of a time series is gaussian or nearly so the method of proof used here is based on approximations of absolutely regular and strong mixing random variables rvs by independent rvs

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

multivariate quantiles and multiple output regression quantiles from l1 optimization to halfspace depth a new multivariate concept of quantile based on a directional version of koenker and bassett s traditional regression quantiles is introduced for multivariate location and multiple output regression problems in their empirical version those quantiles can be computed efficiently via linear programming techniques consistency bahadur representation and asymptotic normality results are established most importantly the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of tukey this relation does not only allow for efficient depth contour computations by means of parametric linear programming but also for transferring from the quantile to the depth universe such asymptotic results as bahadur representations finally linear programming duality opens the way to promising developments in depth related multivariate rank based inference

recursive kernel density estimation for time series we consider the recursive estimation of the probability density function of continuous random variables from a strongly mixing random sample we revisit here earlier research on this subject by considering a more general class of recursive estimators including the usual ones we derive the quadratic mean error of the considered class of estimators moreover we establish a central limit theorem by using lindeberg s method resulting in a simplification of the existing assumptions on the sequence of smooth parameters and the mixing coefficient this is the main contribution of this paper finally the feasibility of the proposed estimator is illustrated throughout an empirical study

on density estimation from ergodic processes we consider the problem of lp consistent density estimation from the initial segments of strongly dependent processes it is shown that no procedure can consistently estimate the one dimensional marginal density of every stationary ergodic process for which such a density exists a similar result is established for the problem of estimating the support of the marginal distribution of an ergodic process

adaptive density estimation from data with small measurement errors in this paper we study the problem of density estimation from data that contains small measurement errors the only assumption on these errors is that the maximal measurement error is bounded by some real number converging to zero for sample size tending to infinity in particular we do not assume that the measurement errors are independent with expectation zero we estimate the density by a standard kernel density estimate applied to data with measurement errors and derive a data driven method to choose its bandwidth we derive an adaptation result for this estimate and analyze the expected l 1 error of our density estimate depending on the smoothness of the density and the size of the maximal measurement error the results are applied in a density estimation problem in a simulation model where we show under suitable assumptions that the l 1 error of our newly proposed estimate converges to zero much faster than the l 1 error of the standard kernel density estimate if both are based on the same number of observations in the simulation model the performance of the method in case of finite sample size is analyzed using simulated data

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

estimating a function from ergodic samples with additive noise we study the problem of estimating an unknown function from ergodic samples corrupted by additive noise it is shown that one can consistently recover an unknown measurable function in this setting if the one dimensional 1 d distribution of the samples is comparable to a known reference distribution and the noise is independent of the samples and has known mixing rates the estimates are applied to deterministic sampling schemes in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories

divergence estimation of continuous distributions based on data dependent partitions we present a universal estimator of the divergence d p spl par q for two arbitrary continuous distributions p and q satisfying certain regularity conditions this algorithm which observes independent and identically distributed i i d samples from both p and q is based on the estimation of the radon nikodym derivative dp dq via a data dependent partition of the observation space strong convergence of this estimator is proved with an empirically equivalent segmentation of the space this basic estimator is further improved by adaptive partitioning schemes and by bias correction the application of the algorithms to data with memory is also investigated in the simulations we compare our estimators with the direct plug in estimator and estimators based on other partitioning approaches experimental results show that our methods achieve the best convergence performance in most of the tested cases

a real schur form method for modeling singularly perturbed systems a method is presented for modeling a two time scale system in the singularly perturbed form the method uses an ordered real schur decomposition which can be efficiently computed using standard subroutines from eispack three results are given first it is shown that any two time scale system can be modeled in the singularly perturbed form by a transformation into an ordered real schur form followed by balancing second under some conditions on the ordered real schur decomposition a procedure is given to achieve the modeling task with all fast variables chosen from the original state variables third necessary and sufficient conditions are given to achieve modeling by permutation of the original state variables

uniformly root n consistent density estimators for weakly dependent invertible linear processes convergence rates of kernel density estimators for stationary time series are well studied for invertible linear processes we construct a new density estimator that converges in the supremum norm at the better parametric rate n 1 2 our estimator is a convolution of two different residual based kernel estimators we obtain in particular convergence rates for such residual based kernel estimators these results are of independent interest

decomposition of two time scale linear systems via the relevant eigenspace this paper presents a very reliable and computationally simple algorithm to transform a two time scale system in the singularly perturbed form by permutation followed by scaling assuming that certain conditions are satisfied the algorithm is based on computing orthonormal bases which span the left and the right invariant subspaces associated with the slow eigenvalues the algorithm does not require the system to be diagonalizable moreover the algorithm works even if the original system is badly scaled examples are given to illustrate the proposed algorithm

universal denoising of discrete time continuous amplitude signals we consider the problem of reconstructing a discrete time signal sequence with continuous valued components corrupted by a known memoryless channel when performance is measured using a per symbol loss function satisfying mild regularity conditions we develop a sequence of denoisers that although independent of the distribution of the underlying ldquocleanrdquo sequence is universally optimal in the limit of large sequence length this sequence of denoisers is universal in the sense of performing as well as any sliding window denoising scheme which may be optimized for the underlying clean signal our results are initially developed in a ldquosemi stochasticrdquo setting where the noiseless signal is an unknown individual sequence and the only source of randomness is due to the channel noise it is subsequently shown that in the fully stochastic setting where the noiseless sequence is a stationary stochastic process our schemes universally attain optimum performance the proposed schemes draw from nonparametric density estimation techniques and are practically implementable we demonstrate efficacy of the proposed schemes in denoising gray scale images in the conventional additive white gaussian noise awgn setting with additional promising results for less conventional noise distributions

the l sub 1 and l sub 2 strong consistency of recursive kernel density estimation from dependent samples the l sub 1 and l sub 2 strong consistency of recursive kernel density estimators is established for mixing and ergodic stationary processes sharp rates of almost sure convergence are obtained in the l sub 2 case for mixing processes in addition the concept and properties of hilbert space valued mixingales are developed and strong laws of large numbers are given

recursive nonparametric estimation for time series this paper considers online kernel estimation for both short and long range dependent time series data utilizing the predictive dependence measure of wu we carefully study the asymptotic properties of recursive kernel density and regression estimators for a general class of stationary processes in particular we prove that the proposed estimators have the asymptotic normality and the corresponding central limit theorems are provided in addition we establish the sharp laws of the iterated logarithms that precisely characterize the asymptotic almost sure behavior of the proposed estimators

stability bounds of singularly perturbed discrete systems the authors propose a systematic approach to determine the exact stability bound of singularly perturbed discrete time systems by combining three conditions of the critical stability criterion with a bialternate product for discrete time systems the stability problems of two types of singularly perturbed discrete time systems can be solved via the derived scheme several examples are given to illustrate the proposed results

rates of convergence and optimal spectral bandwidth for long range dependence for a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like 1 2h as 0 for 1 2 h 1 apart from a slowly varying factor which may be of unknown form we consider a discrete average of the periodogram across the frequencies 2 j n j 1 m wherem andm n 0 asn we study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates and deduce an optimal choice ofm

quantile regression methods for recursive structural equation models abstract two classes of quantile regression estimation methods for the recursive structural equation models of chesher 2003 identification in nonseparable models econometrica 71 1405 1441 are investigated a class of weighted average derivative estimators based directly on the identification strategy of chesher is contrasted with a new control variate estimation method the latter imposes stronger restrictions achieving an asymptotic efficiency bound with respect to the former class an application of the methods to the study of the effect of class size on the performance of dutch primary school students shows that i reductions in class size are beneficial for good students in language and for weaker students in mathematics ii larger classes appear beneficial for weaker language students and iii the impact of class size on both mean and median performance is negligible

robust stability of singularly perturbed discrete delay systems this paper presents new sufficient conditions for the robust stability of singularly perturbed discrete systems subjected to multiple delayed perturbations for a given spl epsi spl isin 0 spl epsi sub 0 where spl epsi sub 0

stabilization bound of discrete two time scale systems abstract in this paper the exact bound problem in various types of discrete two time scale tts system is considered a set of stability criteria based on the frequency domain representation is derived two interesting examples are used to demonstrate the proposed scheme

optimal control of a particular class of singularly perturbed nonlinear discrete time systems this note studies a class of discrete time nonlinear systems which depend on a small parameter using the singular perturbation theory in a systematic way we give a trajectory approximation result based on the decomposition of the model into reduced and boundary layer models this decomposition is used to analyze optimal control via maximum principle of such systems as a result significant order reduction of optimal control problems is achieved

nonparametric estimation of conditional distributions estimation of conditional distributions is considered it is assumed that the conditional distribution is either discrete or that it has a density with respect to the lebesgue measure partitioning estimates of the conditional distribution are constructed and results concerning consistency and rate of convergence of the integrated total variation error of the estimates are presented

distribution free consistency of a nonparametric kernel regression estimate and classification it is shown that the kernel estimate of the regression e y x x is weakly or strongly consistent for almost all x mu where mu is the probability measure of x the result is valid for any distribution of x the asymptotical optimality of classification rules derived from the estimate is examined the optimality is independent of class distributions i e it is distribution free

an approach to multivariate covariate dependent quantile contours with application to bivariate conditional growth charts multivariate quantile contours are useful in numerous applications and have been studied in different contexts however no easy solutions exist when dynamic and conditional quantile contours are needed without strong distributional assumptions in this article we propose a new form of bivariate quantile contours and a two stage estimation procedure to take time effect into account the proposed procedure relies on quantile regression for longitudinal data and is flexible to include potentially important covariates as necessary in addition we propose a visual model assessment tool and discuss a practical guideline for model selection the performance of the proposed methodology is demonstrated by a simulation study as well as an application to joint height weight screening of young children in the united states we construct bivariate growth charts by a nested sequence of age dependent and covariate varying quantile contours of height and weight and use it to locate an individual subject s percentile

inference on multivariate m estimators based on bivariate censored data consider two random variables subject to random right censoring for example the survival times of twins or two recurrence times of a certain disease recorded on the same individual of interest is the estimation of a multivariate m functional of these two random variables asymptotic results for the proposed estimator are established under general conditions on the criterion function of the m estimator the obtained results include the asymptotic normality of the estimator its local power and construction of a confidence region our primary example of interest is the bivariate l1 median we compare the small sample behavior of this estimator with that of the vector of marginal medians by means of a simulation study finally we consider a dataset on kidney dialysis patients and estimate the median time to two different infections for these individuals

convergence rates in density estimation for data from infinite order moving average processes the effect of long range dependence in nonparametric probability density estimation is investigated under the assumption that the observed data are a sample from a stationary infinite order moving average process it is shown that to first order the mean integrated squared error mise of a kernel estimator for moving average data may be expanded as the sum of mise of the kernel estimator for a same sizerandom sample plus a term proportional to the variance of the moving average sample mean the latter term does not depend on bandwidth and so imposes a ceiling on the convergence rate of a kernel estimator regardless of how bandwidth is chosen this ceiling can be quite significant in the case of long range dependence we show thatall density estimators have the convergence rate ceiling possessed by kernel estimators

simplistic universal coding a conceptually simple coding method may be described as follows the source sequence is parsed into fixed length blocks and a list of these blocks is placed in a dictionary in the lossless case the dictionary is transmitted and each successive block is encoded by giving its dictionary location in the lossy case the smallest collection of blocks such that every member of the dictionary is within distortion spl delta of the collection is determined this codebook is transmitted and each successive block is encoded by giving the location of a member of the code within spl delta of it we show that by optimizing on the block length this method is universal that is for any ergodic process it achieves entropy in the limit in the lossless case and the rate distortion function r spl delta in the lossy case

on fractile transformation of covariates in regression the need for comparing two regression functions arises frequently in statistical applications comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different for the populations for instance in econometric studies the prices of commodities and people s incomes observed at different time points may not be on comparable scales due to inflation and other economic factors in this article we describe a method of standardizing the covariates and estimating the transformed regression function which then become comparable we develop smooth estimates of the fractile regression function and study its statistical properties analytically as well as numerically we also provide a few real examples that illustrate the difficulty in comparing the usual regression functions and motivate the need for the fractile transformation our analysis of the real examples leads to new and useful statistical conclusions that are missed b

identification in nonseparable models weak nonparametric restrictions are developed sufficient to identify the values of derivatives of structural functions in which latent random variables are nonseparable these derivatives can exhibit stochastic variation in a microeconometric context this allows the impact of a policy intervention as measured by the value of a structural derivative to vary across people who are identical as measured by covariates when the restrictions are satisfied quantiles of the distribution of a policy impact across people can be identified the identification restrictions are local in the sense that they are specific to the values of the covariates and the specific quantiles of latent variables at which identification is sought the conditions do not include the commonly required independence of latent variables and covariates they include local versions of the classical rank and order conditions and local quantile insensitivity conditions values of structural derivatives are identified by functionals of quantile regression functions and can be estimated using the same functionals applied to estimated quantile regression functions

limits to consistent on line forecasting for ergodic time series this article concerns problems of time series forecasting under the weakest of assumptions related results are surveyed and are points of departure for the developments here some of which are new and others are new derivations of previous findings the contributions in this study are all negative showing that various plausible prediction problems are unsolvable or in other cases are not solvable by predictors which are known to be consistent when mixing conditions hold

on arbitrarily slow rates of global convergence in density estimation let a density f on rd be estimated by fn x x1 xn where x rd fn is a borel measurable function of its arguments and x1 xn are independent random vectors with common density f let p 1 be a constant one of the main results of this note is that for every sequence fn and for every positive number sequence an satisfying lim an 0 there exists an f such that e left smallint f n left x right f left x right p dx right a n infinitely often

m estimation convexity and quantiles the paper develops a class of extensions of the univariate quantile function to the multivariate case m quantiles related in a certain way to m parameters of a probability distribution and their m estimators the spatial geometric quantiles recently introduced by koltchinskii and dudley and by chaudhuri as well as the regression quantiles of koenker and basset are the examples of the m quantile function discussed in the paper we study the main properties of m quantiles and develop the asymptotic theory of empirical m quantiles we use m quantiles to extend l parameters and l estimators to the multivariate case to introduce a bootstrap test for spherical symmetry of a multivariate distribution and to extend the notion of regression quantiles to multiresponse linear regression models

determining the complexity of fh ss sequence by approximate entropy high complexity of frequency hopping fh spreadspectrum ss sequence is of great importance to high security multiple access communication systems for it makes fh ss sequence difficult to be analyzed with the growing development in the design of fh ss sequence in much wider fields the wellknown complexity measuresiquestthe linear complexity lc the linear complexity profile lcp and the k error linear complexity k error lc iquestare widely used but not sufficient to evaluate the complexities of the sequences available such as the cryptographical sequence and the chaotic sequence families in this paper a new complexity metric to evaluate the unpredictability of fh ss sequence based on the approximate entropy apen is proposed in the view of the maximal randomness of the sequences with arbitrary length and the theoretical bounds of the apen are derived from a probabilistic point of view simulations and analysis results show that the proposed apen works effectively to discern the changing complexities of the fh ss sequences with small number of samples which provide superior performance over its candidates

how sampling reveals a process a series of observations xi 1 xi 2 xi 3 ldots is presented to us and at each time n when we have observed the first n of them we are called upon to give our guess for what stochastic process produced the data a universal scheme is given which for any bernoulli process not necessarily independent gives a sequence of processes that converges in a strong sense the bar d metric to the real process in addition to this main result many others are given which put it into proper perspective in particular it is shown that in a certain sense the class of bernoulli processes is the largest one for which such a universal scheme is possible

function estimation via wavelet shrinkage for long memory data in this article we study function estimation via wavelet shrinkage for data with long range dependence we propose a fractional gaussian noise model to approximate nonparametric regression with long range dependence and establish asymptotics for minimax risks because of long range dependence the minimax risk and the minimax linear risk converge to 0 at rates that differ from those for data with independence or short range dependence wavelet estimates with best selection of resolution leveldependent threshold achieve minimax rates over a wide range of spaces cross validation for dependent data is proposed to select the optimal threshold the wavelet estimates significantly outperform linear estimates the key to proving the asymptotic results is a wavelet vaguelette decomposition which decorrelates fractional gaussian noise such wavelet vaguelette decomposition is also very useful in fractal signal processing

the rates of convergence of kernel regression estimates and classification rules both nonrecursive and recursive nonparametric regression estimates are studied the rates of weak and strong convergence of kernel estimates as well as corresponding multiple classification errors are derived without assuming the existence of the density of the measurements an application of the obtained results to a nonparametric bayes predication is presented

hilbert space embeddings and metrics on probability measures a hilbert space embedding for probability measures has recently been proposed with applications including dimensionality reduction homogeneity testing and independence testing this embedding represents any probability measure as a mean element in a reproducing kernel hilbert space rkhs a pseudometric on the space of probability measures can be defined as the distance between distribution embeddings we denote this as k indexed by the kernel function k that defines the inner product in the rkhs we present three theoretical properties of k first we consider the question of determining the conditions on the kernel k for which k is a metric such k are denoted characteristic kernels unlike pseudometrics a metric is zero only when two distributions coincide thus ensuring the rkhs embedding maps all distributions uniquely i e the embedding is injective while previously published conditions may apply only in restricted circumstances e g on compact domains and are difficult to check our conditions are straightforward and intuitive integrally strictly positive definite kernels are characteristic alternatively if a bounded continuous kernel is translation invariant on d then it is characteristic if and only if the support of its fourier transform is the entire d second we show that the distance between distributions under k results from an interplay between the properties of the kernel and the distributions by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies third to understand the nature of the topology induced by k we relate k to other popular metrics on probability measures and present conditions on the kernel k under which k metrizes the weak topology

compression based methods for nonparametric prediction and estimation of some characteristics of time series we address the problem of online prediction for time series we show that any universal code or a universal data compressor can be used as a basis for constructing asymptotically optimal methods for this problem for a certain class of stationary and ergodic processes