the gaussian measure of shifted balls let be a centered gaussian measure on a hilbert spaceh and let b r subseteq h be the centered ball of radiusr 0 fora h and mathop lim limits t mathbf to mathbf infty mathbf r t t mathbf a we give the exact asymptotics of b r t t a ast also upper and lower bounds are given when is defined on an arbitrary separable banach space our results range from small deviation estimates to large deviation estimates

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

convergence rates for posterior distributions and adaptive estimation the goal of this paper is to provide theorems on convergence rates of posterior distributions that can be applied to obtain good convergence rates in the context of density estimation as well as regression we show how to choose priors so that the posterior distributions converge at the optimal rate without prior knowledge of the degree of smoothness of the density function or the regression function to be estimated

digital audio signal coding method through allocation of quantization bits to sub band samples split from the audio signal disclosed is an invention concerning a quantization bit number allocating section for sub band coding wherein data important for human auditory sense are efficiently coded within a limited coding bit capacity to provide a high quality digital audio signal the quantization bit number allocating section comprises a level calculating section a logarithm calculating section an index calculating section a quantization bit number calculating section a logarithm weighting table and a sub band weighting table wherein the quantization bit number is determined every prescribed time according to human auditory sense and the characteristic of an input digital audio signal

the logistic normal distribution for bayesian nonparametric predictive densities abstract this article models the common density of an exchangeable sequence of observations by a generalization of the process derived from a logistic transform of a gaussian process the support of the logistic normal includes all distributions that are absolutely continuous with respect to the dominating measure of the observations the logistic normal family is closed in the prior to posterior bayes analysis with the observations entering the posterior distribution through the covariance function of the gaussian process the covariance of the gaussian process plays the role of a smoothing kernel three features of the model provide a flexible structure for computing the predictive density a the mean of the gaussian process corresponds to the prior mean of the random density b the prior variance of the gaussian process controls the influence of the data in the posterior process as the variance increases the predictive density has greater fidelity to the data c the prior covariance of the gau

density estimation by wavelet thresholding density estimation is a commonly used test case for nonparametric estimation methods we explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients minimax rates of convergence are studied over a large range of besov function classes b pq and for a range of global l p error measures 1 p p some form of nonlinearity is essential since the minimax linear estimators are suboptimal by polynomial powers of n a second approach using an approximation of a gaussian white noise model in a mallows metric is used to attain exactly optimal rates of convergence for quadratic error p 2

adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth we consider nonparametric bayesian estimation inference using a rescaled smooth gaussian field as a prior for a multidimensional function the rescaling is achieved using a gamma variable and the procedure can be viewed as choosing an inverse gamma bandwidth the procedure is studied from a frequentist perspective in three statistical settings involving replicated observations density estimation regression and classification we prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax optimal up to a logarithmic factor whatever the regularity level of the data generating distribution thus the hierachical bayesian procedure with a fixed prior is shown to be fully adaptive

rates of contraction of posterior distributions based on gaussian process priors we derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on gaussian processes the rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel hilbert space of the gaussian process and the small ball probabilities of the gaussian process we determine these quantities for a range of examples of gaussian priors and in several statistical settings for instance we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a fractionally integrated brownian motion we also consider regression with gaussian errors and smooth classification under a logistic or probit link function combined with various priors

an improved tau method for the multi dimensional fractional rayleigh stokes problem for a heated generalized second grade fluid abstract we develop efficient algorithms based on the legendre tau approximation for one and two dimensional fractional rayleigh stokes problems for a generalized second grade fluid the time fractional derivative is described in the riemann liouville sense discussions on the l 2 convergence of the proposed method are presented numerical results for one and two dimensional examples with smooth and nonsmooth solutions are provided to verify the validity of the theoretical analysis and to illustrate the efficiency of the proposed algorithms

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

approximation metric entropy and small ball estimates for gaussian measures a precise link proved by kuelbs and li relates the small ball behavior of a gaussian measure on a banach space e with the metric entropy behavior of k the unit ball of the reproducing kernel hilbert space of in e we remove the main regularity assumption imposed on the unknown function in the link this enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete gaussian processes moreover we show that the small ball behavior of a gaussian process is also tightly connected with the speed of approximation by finite rank processes

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

metric entropy and the small ball problem for gaussian measures abstract we establish a precise link between the small ball problem for a gaussian measure on a separable banach space and the metric entropy of the unit ball of the hubert space h generating this link allows us to compute small ball probabilities from metric entropy results and vice versa

towards a practicable bayesian nonparametric density estimator summary nonparametric density estimators smooth the empirical distribution function and are sensitive to the choice of smoothing parameters this paper develops an hierarchical bayes formulation for the smoothing problem the prior distribution for the density function is the logistic normal process which is a logistic transform of a gaussian process the covariance of the gaussian process is a smoothing kernel and has parameters that control the degree of smoothness the likelihood function for the smoothing parameters and their posterior density are computed from an approximation of the joint moments of the logistic normal process the marginal predictive density mixes the conditional predictive density given the smoothing parameters with their posterior distribution this hierarchical bayes analysis provides a fully automated data dependent method for smoothing and selects the amount of smoothing that is coherent with its prior specification a bayesian approach to nonparametric problems is to construct a probability distribution on the relevant space of distribution functions and to update this distribution as observations become available families of prior distributions must not only have stochastic properties that can adequately model prior information but also must provide tractable models the dirichlet process ferguson 1973 has been quite successful on both counts however it is not directly applicable to absolutely continuous distributions because the posterior probability of the space of absolutely continuous distributions is zero lo 1984 convolves the dirichlet process with a smoothing kernel to model a density function the computational complexity of the posterior mean increases exponentially with the sample size an alternative to the dirichlet model is provided by the logistic normal process

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

bayesian regression with nonparametric heteroskedasticity abstract this paper studies large sample properties of a semiparametric bayesian approach to inference in a linear regression model the approach is to model the distribution of the regression error term by a normal distribution with the variance that is a flexible function of covariates the main result of the paper is a semiparametric bernstein von mises theorem under misspecification even when the distribution of the regression error term is not normal the posterior distribution of the properly recentered and rescaled regression coefficients converges to a normal distribution with the zero mean and the variance equal to the semiparametric efficiency bound

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

joint estimation of quantile planes over arbitrary predictor spaces abstractin spite of the recent surge of interest in quantile regression joint estimation of linear quantile planes remains a great challenge in statistics and econometrics we propose a novel parameterization that characterizes any collection of noncrossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar vector and function valued parameters statistical models based on this parameterization inherit a fast computation of the likelihood function enabling penalized likelihood or bayesian approaches to model fitting we introduce a complete bayesian methodology by using gaussian process prior distributions on the function valued parameters and develop a robust and efficient markov chain monte carlo parameter estimation the resulting method is shown to offer posterior consistency under mild tail and regularity conditions we present several illustrative examples where the new method is compared against existing approaches and is found to o