local asymptotics for polynomial spline regression in this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem the polynomial spline spaces we consider include univariate splines tensor product splines and bivariate or multivariate splines on triangulations we establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation the asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities error distributions and regression functions the bias is controlled by the minimum l norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate the control of bias relies on the stability in l norm of l 2 projections onto polynomial spline spaces asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory in addition a preliminary analysis of additive models is provided

partially monotone tensor spline estimation of the joint distribution function with bivariate current status data the analysis of the joint cumulative distribution function cdf with bivariate event time data is a challenging problem both theoretically and numerically this paper develops a tensor spline based sieve maximum likelihood estimation method to estimate the joint cdf with bivariate current status data the i splines are used to approximate the joint cdf in order to simplify the numerical computation of a constrained maximum likelihood estimation problem the generalized gradient projection algorithm is used to compute the constrained optimization problem based on the properties of b spline basis functions it is shown that the proposed tensor spline based nonparametric sieve maximum likelihood estimator is consistent with a rate of convergence potentially better than n 1 3 under some mild regularity conditions the simulation studies with moderate sample sizes are carried out to demonstrate that the finite sample performance of the proposed estimator is generally satisfactory

estimation and variable selection for generalized additive partial linear models we study generalized additive partial linear models proposing the use of polynomial spline smoothing for estimation of nonparametric functions and deriving quasi likelihood based estimators for the linear parameters we establish asymptotic normality for the estimators of the parametric components the procedure avoids solving large systems of equations as in kernel based procedures and thus results in gains in computational simplicity we further develop a class of variable selection procedures for the linear parameters by employing a nonconcave penalized quasi likelihood which is shown to have an asymptotic oracle property monte carlo simulations and an empirical example are presented for illustration

rate of convergence in density estimation using neural networks given n i i d observations xi ni 1 taking values in a compact subset of rd such that p denotes their common probability density function we estimate p from an exponential family of densities based on single hidden layer sigmoidal networks using a certain minimum complexity density estimation scheme assuming that p possesses a certain exponential representation we establish a rate of convergence independent of the dimension d for the expected hellinger distance between the proposed minimum complexity density estimator and the true underlying density p

analysis of additive dependencies and concurvities using smallest additive principal components additive principal components are a nonlinear generalization of linear principal components their distinguishing feature is that linear forms i a i x i are replaced with additive functions i f i x i a considerable amount of flexibility for fitting data is gained when linear methods are replaced with additive ones our interest is in the smallest principal components which is somewhat uncommon smallest additive principal components amount to data descriptions in terms of approximate implicit equations i f i x i 0 estimation of such equations is a data analytic method in its own right competing in some cases with the more customary regression approaches it is also a diagnostic tool in additive regression for detection of so called concurvity this term describes degeneracies among predictor variables in additive regression models similar to collinearity in linear regression models concurvity may lead to statistically unstable contributions of variables to additive models as an example we show in a reanalysis of the ozone data from breiman and friedman that concurvity does indeed exist in this particular data a fact which may impact the interpretation of the additive fits in the second half of this paper we give some second order theory including the description of null situations and eigenexpansions derived from associated eigenproblems we show how ace and additive principal components are related and we outline some analytical methods for theoretical calculations of additive principal components lastly we consider methods of estimation and computation additive principal components have had a long tradition in psychometric research and correspondence analysis they have started receiving attention by statisticians only in recent years

the parisi formula has a unique minimizer in 1979 parisi phys rev lett 43 1754 1756 1979 predicted a variational formula for the thermodynamic limit of the free energy in the sherrington kirkpatrick model and described the role played by its minimizer this formula was verified in the seminal work of talagrand ann math 163 1 221 263 2006 and later generalized to the mixed p spin models by panchenko ann probab 42 3 946 958 2014 in this paper we prove that the minimizer in parisi s formula is unique at any temperature and external field by establishing the strict convexity of the parisi functional

projection estimation in multiple regression with application to functional anova models a general theory on rates of convergence of the least squares projection estimate in multiple regression is developed the theory is applied to the functional anova model where the multivariate regression function is modeled as a specified sum of a constant term main effects functions of one variable and selected interaction terms functions of two or more variables the least squares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed anova structure of the regression function the linear spaces that serve as building blocks can be any of the ones commonly used in practice polynomials trigonometric polynomials splines wavelets and finite elements the rate of convergence result that is obtained reinforces the intuition that low order anova modeling can achieve dimension reduction and thus overcome the curse of dimensionality moreover the components of the projection estimate in an appropriately defined anova decomposition provide consistent estimates of the corresponding components of the regression function when the regression function does not satisfy the assumed anova form the projection estimate converges to its best approximation of that form

on properties of parisi measures we investigate the structure of parisi measures the functional order parameters of mixed p spin models in mean field spin glasses in the absence of external field we prove that a parisi measure satisfies the following properties first at all temperatures the support of any parisi measure contains the origin if it contains an open interval then the measure has a smooth density on this interval next we give a criterion on temperature parameters for which a parisi measure is neither replica symmetric nor one replica symmetry breaking finally we show that in the sherrington kirkpatrick model slightly above the critical temperature the largest number in the support of a parisi measure is a jump discontinuity an analogue of these results is discussed in the spherical mixed p spin models as a tool to establish these facts and of independent interest we study functionals of the associated parisi pdes and derive regularity properties of their solutions

on deep learning as a remedy for the curse of dimensionality in nonparametric regression assuming that a smoothness condition and a suitable restriction on the structure of the regression function hold it is shown that least squares estimates based on multilayer feedforward neural networks are able to circumvent the curse of dimensionality in nonparametric regression the proof is based on new approximation results concerning multilayer feedforward neural networks with bounded weights and a bounded number of hidden neurons the estimates are compared with various other approaches by using simulated data

minimax nonparametric classification ii model selection for adaptation for pt i see ibid vol 45 no 7 p 2271 84 1999 we study nonparametric estimation of a conditional probability for classification based on a collection of finite dimensional models for the sake of flexibility different types of models linear or nonlinear are allowed as long as each satisfies a dimensionality assumption we show that with a suitable model selection criterion the penalized maximum likelihood estimator has a risk bounded by an index of resolvability expressing a good tradeoff among approximation error estimation error and model complexity the bound does not require any assumption on the target conditional probability and can be used to demonstrate the adaptivity of estimators based on model selection examples are given with both splines and neural nets and problems of high dimensional estimation are considered the resulting adaptive estimator is shown to behave optimally or near optimally over sobolev classes with unknown orders of interaction and smoothness and classes of integrable fourier transform of gradient in terms of rates of convergence the performance is the same as if one knew which of them contains the true conditional probability in advance the corresponding classifier also converges optimally or nearly optimally simultaneously over these classes

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

how fast are the particles of super brownian motion in this paper we investigate fast particles in the range and support ofsuper brownian motion in the historical setting in this setting eachparticle of super brownian motion alive at time t is represented by apath w 0 t d and the state of historical super brownian motionis a measure on the set of paths typical particles have brownian paths however in the uncountable collection of particles in the range of asuper brownian motion there are some which at exceptional times movefaster than brownian motion we determine the maximal speed of allparticles during a given time period e which turns out to be afunction of the packing dimension of e a path w in the support ofhistorical super brownian motion at time t is called a fast if wecalculate the hausdorff dimension of the set of a fast paths in thesupport and the range of historical super brownian motion a valuabletool in the proofs is a uniform dimension formula for the browniansnake which reduces dimension problems in the space of stopped paths to dimension problems on the line

improving point and interval estimators of monotone functions by rearrangement suppose that a target function is monotonic and an available original estimate of this target function is not monotonic rearrangements univariate and multivariate transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function furthermore suppose an original confidence interval which covers the target function with probability at least 1 is defined by an upper and lower endpoint functions that are not monotonic then the rearranged confidence interval defined by the rearranged upper and lower endpoint functions is monotonic shorter in length in common norms than the original interval and covers the target function with probability at least 1 we illustrate the results with a growth chart example

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

spline backfitted kernel smoothing of nonlinear additive autoregression model application of nonparametric and semiparametric regression techniques to high dimensional time series data has been hampered due to the lack of effective tools to address the curse of dimensionality under rather weak conditions we propose spline backfitted kernel estimators of the component functions for the nonlinear additive time series data that are both computationally expedient so they are usable for analyzing very high dimensional time series and theoretically reliable so inference can be made on the component functions with confidence simulation experiments have provided strong evidence that corroborates the asymptotic theory

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

identification of non linear additive autoregressive models summary we propose a lag selection method for non linear additive autoregressive models that is based on spline estimation and the bayes information criterion the additive structure of the autoregression function is used to overcome the curse of dimensionality whereas the spline estimators effectively take into account such a structure in estimation a stepwise procedure is suggested to implement the method proposed a comprehensive monte carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local polynomial based methods consistency of the lag selection method based on the bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing which provides the first theoretical result of this kind for spline smoothing of weakly dependent data

on the energy landscape of the mixed even p spin model we investigate the energy landscape of the mixed even p spin model with ising spin configurations we show that for any given energy level between zero and the maximal energy with overwhelming probability there exist exponentially many distinct spin configurations such that their energies stay near this energy level furthermore their magnetizations and overlaps are concentrated around some fixed constants in particular at the level of maximal energy we prove that the hamiltonian exhibits exponentially many orthogonal peaks this improves the results of chatterjee disorder chaos and multiple valleys in spin glasses 2009 and ding et al ann probab 43 6 3468 3493 2015 where the former established a logarithmic size of the number of the orthogonal peaks while the latter proved a polynomial size our second main result obtains disorder chaos at zero temperature and at any external field as a byproduct this implies that the fluctuation of the maximal energy is superconcentrated when the external field vanishes and obeys a gaussian limit law when the external field is present

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

smooth function approximation using neural networks an algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented in this paper the approach is implemented for the approximation of smooth batch data containing the function s input output and possibly gradient information the training set is associated to the network adjustable parameters by nonlinear weight equations the cascade structure of these equations reveals that they can be treated as sets of linear systems hence the training process and the network approximation properties can be investigated via linear algebra four algorithms are developed to achieve exact or approximate matching of input output and or gradient based training sets their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques

functional anova modeling for proportional hazards regression the logarithm of the relative risk function in a proportional hazards model involving one or more possibly time dependent covariates is treated as a specified sum of a constant term main effects and selected interaction terms maximum partial likelihood estimation is used where the maximization is taken over a suitably chosen finite dimensional estimation space whose dimension increases with the sample size and which is constructed from linear spaces of functions of one covariate and their tensor products the l 2 rate of convergence for the estimate and its anova components is obtained an adaptive numerical implementation is discussed whose performance is compared to full likelihood hazard regression both with and without the restriction to proportional hazards

oracally efficient two step estimation of generalized additive model the generalized additive model gam is a multivariate nonparametric regression tool for non gaussian responses including binary and count data we propose a spline backfitted kernel sbk estimator for the component functions and the constant which are oracally efficient under weak dependence the sbk technique is both computationally expedient and theoretically reliable thus usable for analyzing high dimensional time series inference can be made on component functions based on asymptotic normality simulation evidence strongly corroborates the asymptotic theory the method is applied to estimate insolvent probability and to obtain higher accuracy ratio than a previous study supplementary materials for this article are available online

polynomial splines and their tensor products in extended linear modeling 1994 wald memorial lecture analysis of variance type models are considered for a regression function or for the logarithm of a probability function conditional probability function density function conditional density function hazard function conditional hazard function or spectral density function polynomial splines are used to model the main effects and their tensor products are used to model any interaction components that are included in the special context of survival analysis the baseline hazard function is modeled and nonproportionality is allowed in general the theory involves the l 2 rate of convergence for the fitted model and its components the methodology involves least squares and maximum likelihood estimation stepwise addition of basis functions using rao statistics stepwise deletion using wald statistics and model selection using the bayesian information criterion cross validation or an independent test set publicly available software written in c and interfaced to s s plus is used to apply this methodology to real data

log density estimation in linear inverse problems we estimate a probability density function p which is related by a linear operator k to a density function q in sequences of regular exponential families based on a random sample from q in this paper deconvolution and positron emission tomography are considered the logarithm of the density function is approximated by basis functions consisting of singular functions of k while direct maximum likelihood or minimum kullback leibler density estimation in exponential families selects the parameters to match the moments of the basis functions to the sample moments in the inverse problem the moment of each singular function is related to a corresponding moment of the direct problem by a factor given by a singular value a of k thus an appropriate analogue of the maximum likelihood estimate is obtained by matching moments with respect to p to 1 a times the empirical moments associated with the sample from q bounds on the kullback leibler distance between the true density and the estimators are obtained and rates of convergence are established for log density functions having a measure of smoothness the density estimator converges to the unknown density in the kullback leibler sense and in the l2 sense at a rate determined not only by the order of smoothness of the log density and the dimension of data but also by the decay rate of the singular values of the operator a minimax lower bound for deconvolution is provided under certain conditions numerical examples using simulated data are provided to illustrate the finite sample performance of the proposed method for deconvolution and positron emission tomography

using specially designed exponential families for density estimation i this paper have y being portions of the real line or of the plane but the methodology applies just as well to higher dimensionalities and to more complicated spaces estimates of g y are traditionally constructed in two quite different ways by maximum likelihood fitting within some parametric family such as the normal or by nonparametric methods such as kernel density estimation these two methods can be combined by putting an exponential family through a nonparametric estimator the resulting hybrid estimators are the specially designed exponential families of the title figure 1 shows a simple example of this methodology the y are pain i scores for n s 67 women each obtained by averaging the results from a questionnaire administered after an operation the scale runs from y s 0 s w x

fluctuations of the free energy in the mixed p spin models with external field we show that the free energy in the mixed p spin models of spin glasses does not superconcentrate in the presence of external field which means that its variance is of the order suggested by the poincare inequality this complements the result of chatterjee who showed that the free energy superconcentrates when there is no external field for models without odd p spin interactions for p geqslant 3 we prove the central limit theorem for the free energy at any temperature and give an explicit formula for the limiting variance although we only deal with the case of ising spins all our results can be extended to the spherical models as well

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

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

improving estimates of monotone functions by rearrangement suppose that a target function f0 r r is monotonic namely weakly increasing and an original estimate f of the target function is available which is not weakly increasing many common estimation methods used in statistics produce such estimates f we show that these estimates can always be improved with no harm using rearrangement techniques the rearrangement methods univariate and multivariate transform the original estimate to a monotonic estimate f and the resulting estimate is closer to the true curve f0 in common metrics than the original estimate f we illustrate the results with a computational example and an empirical example dealing with age height growth charts

flexible modeling via a hybrid estimation scheme in generalized mixed models for longitudinal data to circumvent the computational complexity of likelihood inference in generalized mixed models that assume linear or more general additive regression models of covariate effects laplace s approximations to multiple integrals in the likelihood have been commonly used without addressing the issue of adequacy of the approximations for individuals with sparse observations in this article we propose a hybrid estimation scheme to address this issue the likelihoods for subjects with sparse observations use monte carlo approximations involving importance sampling while laplace s approximation is used for the likelihoods of other subjects that satisfy a certain diagnostic check on the adequacy of laplace s approximation because of its computational tractability the proposed approach allows flexible modeling of covariate effects by using regression splines and model selection procedures for knot and variable selection its computational and statistical advantages are illustrated by simulation and by application to longitudinal data from a fecundity study of fruit flies for which overdispersion is modeled via a double exponential family

visualizing the effects of predictor variables in black box supervised learning models in many supervised learning applications understanding and visualizing the effects of the predictor variables on the predicted response is of paramount importance a shortcoming of black box supervised learning models e g complex trees neural networks boosted trees random forests nearest neighbours local kernel weighted methods and support vector regression in this regard is their lack of interpretability or transparency partial dependence plots which are the most popular approach for visualizing the effects of the predictors with black box supervised learning models can produce erroneous results if the predictors are strongly correlated because they require extrapolation of the response at predictor values that are far outside the multivariate envelope of the training data as an alternative to partial dependence plots we present a new visualization approach that we term accumulated local effects plots which do not require this unreliable extrapolation with correlated predictors moreover accumulated local effects plots are far less computationally expensive than partial dependence plots we also provide an r package aleplot as supplementary material to implement our proposed method

hausdorff dimension of the contours of symmetric additive levy processes let x 1 x n denote n independent symmetric levy processes on r d the corresponding additive levy process is defined as the following n parameter random field on r d mathfrak x t x 1 t 1 cdots x n t n quad t in bf r n khoshnevisan and xiao ann probab 30 1 62 100 2002 have found a necessary and sufficient condition for the zero set mathfrak x 1 0 of mathfrak x to be non trivial with positive probability they also provide bounds for the hausdorff dimension of mathfrak x 1 0 which hold with positive probability in the case that mathfrak x 1 0 can be non void here we prove that the hausdorff dimension of mathfrak x 1 0 is a constant almost surely on the event mathfrak x 1 0 neq varnothing moreover we derive a formula for the said constant this portion of our work extends the well known formulas of horowitz israel j math 6 176 182 1968 and hawkes j lond math soc 8 517 525 1974 both of which hold for one parameter levy processes more generally we prove that for every nonrandom borel set f in 0 n the hausdorff dimension of mathfrak x 1 0 cap f is a constant almost surely on the event mathfrak x 1 0 cap f neq varnothing this constant is computed explicitly in many cases

free energy of the spherical mean field model we compute at any temperature the free energy of the multi p spin spherical model when only terms for p even are considered

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

a bayesian approach to nonparametric bivariate regression abstract this article outlines a general bayesian approach to estimating a bivariate regression function in a nonparametric manner it models the function using a bivariate regression spline basis with many terms binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression the regression function is estimated using an estimate of its posterior mean smoothing over the distribution of these binary indicator variables to make the computations tractable all estimates are obtained using markov chain monte carlo sampling extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data driven bivariate surface estimators prominent in the literature it is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings the first

prospects of nonparametric modeling this article brie y summarizes key developments in nonparametric function estimation over the past three decades it highlights some potentially fruitful areas of future research on nonparametric modeling and its applications

nonparametric residue analysis of dynamic pet data with application to cerebral fdg studies in normals kinetic analysis is used to extract metabolic information from dynamic positron emission tomography pet uptake data the theory of indicator dilutions developed in the seminal work of meier and zierler 1954 provides a probabilistic framework for representation of pet tracer uptake data in terms of a convolution between an arterial input function and a tissue residue the residue is a scaled survival function associated with tracer residence in the tissue nonparametric inference for the residue a deconvolution problem provides a novel approach to kinetic analysis critically one that is not reliant on specific compartmental modeling assumptions a practical computational technique based on regularized cubic b spline approximation of the residence time distribution is proposed nonparametric residue analysis allows formal statistical evaluation of specific parametric models to be considered this analysis needs to properly account for the increased flexibility of the nonparametric estimator the met

local asymptotics for regression splines and confidence regions in this paper we study the local behavior of regression splines in particular explicit expressions for the asymptotic pointwise bias and variance of regression splines are obtained in addition asymptotic normality for regression splines is established leading to the construction of approximate confidence intervals and confidence bands for the regression function

an asymptotic property of model selection criteria probability models are estimated by use of penalized log likelihood criteria related to akaike 1973 information criterion aic and minimum description length mdl the accuracies of the density estimators are shown to be related to the tradeoff between three terms the accuracy of approximation the model dimension and the descriptive complexity of the model classes the asymptotic risk is determined under conditions on the penalty term and is shown to be minimax optimal for some cases as an application we show that the optimal rate of convergence is simultaneously achieved for log densities in sobolev spaces w sub 2 sup s u without knowing the smoothness parameter s and norm parameter u in advance applications to neural network models and sparse density function estimation are also provided

nonparametric estimation of an additive model with a link function this paper describes an estimator of the additive components of a nonparametric additive model with a known link function when the additive components are twice continuously differentiable the estimator is asymptotically normally distributed with a rate of convergence in probability of 2 5 n this is true regardless of the finite dimension of the explanatory variable thus in contrast to the existing asymptotically normal estimator the new estimator has no curse of dimensionality moreover the asymptotic distribution of each additive component is the same as it would be if the other components were known with certainty

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

polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form the cox proportional hazards model usually assumes an exponential form for the dependence of the hazard function on covariate variables however in practice this assumption may be violated and other relative risk forms may be more appropriate in this article we consider the proportional hazards model with an unknown relative risk form issues in model interpretation are addressed we propose a method to estimate the relative risk form and the regression parameters simultaneously by first approximating the logarithm of the relative risk form by a spline and then employing the maximum partial likelihood estimation an iterative alternating optimization procedure is developed for efficient implementation statistical inference of the regression coefficients and of the relative risk form based on parametric asymptotic theory is discussed the proposed methods are illustrated using simulation and an application to the veteran s administration lung cancer data

the legendre structure of the parisi formula we show that the parisi formula of the mixed p spin model is a concave function of the squared inverse temperature this allows us to derive a new expression for the parisi formula that involves the inverse temperature and the parisi measure as legendre conjugate variables

thick points of super brownian motion we determine for a super brownian motion xt t 0 in d d 3 the precise gauge function such that almost surely on survival up to time t open image in new window improving a result of barlow evans and perkins about the most visited sites of super brownian motion we also determine upper and lower bounds for the hausdorff dimension spectrum of thick points refining the multifractal analysis of super brownian motion by taylor and perkins the upper bound conjectured to be sharp involves a constant which can be characterized in terms of the upper tails of the associated equilibrium palm distribution

nonparametric regression based on hierarchical interaction models in this paper we introduce the so called hierarchical interaction models where we assume that the computation of the value of a function m mathbb r d rightarrow mathbb r is done in several layers where in each layer a function of at most d inputs computed by the previous layer is evaluated we investigate two different regression estimates based on polynomial splines and on neural networks and show that if the regression function satisfies a hierarchical interaction model and all occurring functions in the model are smooth the rate of convergence of these estimates depends on d and not on d hence in this case the estimates can achieve good rate of convergence even for large d and are in this sense able to circumvent the so called curse of dimensionality

a variational representation for certain functionals of brownian motion in this paper we show that the variational representation log ee f w inf v e 1 2 int 0 1 parallel v s parallel 2 ds f w int 0 cdot v s ds holds where w is a standard d dimensional brownian motion f is any bounded measurable function that maps c 0 1 mathbb r d into mathbb r and the infimum is over all processes v that are progressively measurable with respect to the augmentation of the filtration generated by w an application is made to a problem concerned with large deviations and an extension to unbounded functions is given