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

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

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

an approximate solution based on jacobi polynomials for time fractional convectiondiffusion equation operational matrix of shifted jacobi polynomials is considered solution of time fractional order convectiondiffusion problem is numerically estimated main problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded a system of nonlinear algebraic equations is achieved by approximating the known and unknown functions with the help of shifted jacobi functions problem can be used extensively in science and engineering as in oil reservoir simulations in this article we present a numerical method to numerically solve a time fractional convectiondiffusion equation our method is based on the operational matrices of shifted jacobi polynomials at first problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded then we approximate the known and unknown functions with the help of shifted jacobi functions a system of nonlinear algebraic equations is obtained finally the unknown coefficients are determined by mathematicatm we implemented the proposed method for several examples that they indicate the high accuracy method it should be noted that this method is generalizable to some appropriate problems

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

some aspects of measurement error in a censored regression model abstract when a linear model with a censored dependent variable is written as a regression model the model is nonlinear in the explanatory variables this means that when the explanatory variables are measured with error estimation by instrumental variables will not lead to consistent estimates of the parameters my approach to the problem involves assuming that the variables measured with error can be represented by a reduced form equation and estimating the parameters of the censored regression model by the least absolute deviations estimator this estimator is based on the median of the distribution of the dependent variable rather than the mean and the median is not affected by censoring in the lower tail hence the estimator is consistent under general conditions i also consider the problem of testing for the presence of measurement error

pairwise difference estimators of censored and truncated regression models abstract this paper proposes a class of estimators of the semiparametric censored regression model under the assumption that the error terms are i i d and independent of the covariates the estimators exploit the fact that for a pair of observations a particular transformation of the censored variables depending upon the parameter vector and both covariate vectors will be identically distributed therefore the difference in the transformed dependent variables will be symmetrically distributed around zero when the transformation is evaluated at the true parameter vector an analogous class of estimators for the truncated regression model includes the bhattacharya chernoff and yang 1983 estimator as a special case the estimators are defined as minimizers of u processes so the large sample theory for classical m estimators is extended to this case conditions are given to ensure root n consistency and asymptotic normality of the estimators and a small scale simulation study for the censored regression estimator suggest that it will be well behaved in finite samples

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

nonparametric censored and truncated regression we propose new estimators of the latent regression function in nonparametric cen sored and truncated regression models our estimators are computationally convenient consisting only of two nonparametric regressions and a univariate integral we establish consistency and asymptotic normality for an implementation based on local linear ker nel estimators an extension permits estimation in the presence of a general form of heteroscedasticity both the function m and the distribution f of the error e are unknown the errors are not assumed to be symmetric this paper provides a simple consistent estimator of m x k for some constant k under an additional tail condition we provide a consistent estimator of m x when m is the conditional mean function of the uncensored population also we show that f can be estimated given m x the proposed estimator is extended to deal with the truncated regression model where yi is only observed when it is not censored we also describe extensions to deal with a general form of heteroskedasticity in which the distribution of e could depend in unknown ways on a subset of elements of x for any continuously distributed element xk of x let mk x am x axk we also provide direct estimators of the derivatives mk x in both the censored and truncated regression models these derivatives are interpretable as the marginal effect of a change in x on the underlying uncensored population they can also be used to test or estimate parametric or semiparametric specifications of m x for example mk x is constant if m x is linear in xk and mk x depends only on xk if m x is additive in a function of xk

estimation of type 3 tobit models using symmetric trimming and pairwise comparisons abstract this paper considers estimation of the semiparametric type 3 tobit model we construct two step estimators that use the ideas of symmetric trimming and pairwise comparisons which have been proposed for the censored regression model by powell 1986 and honore and powell 1994 the estimators are consistent and asymptotically normal under two identifying assumptions conditional symmetry and independence between the errors and regressors the ideas and techniques are also implementable in other selection models that are variations of the basic type 3 tobit model we demonstrate this by applying the proposed method in estimating an empirical model considered by udry 1994

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

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

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

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

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

efficient semiparametric estimation of censored and truncated regressions via a smoothed self consistency equation an asymptotically efficient likelihood based semiparametric estimator is derived for the censored regression tobit model based on a new technique for estimating the density function of the residuals in semiparametric models that involve an underlying partially observed regression smoothing the selfconsistency equation for the nonparametric maximum likelihood estimator of the distribution of the residuals yields an integral equation which in some cases can be solved explicitly the resulting estimated density is smooth enough to be used in a practical implementation of the profile likelihood estimator but is still sufficiently close to the nonparametric maximum likelihood estimator to allow estimation of the semiparametric efficient score the parameter estimates obtained by solving the estimated score equations are then asymptotically efficient details of the method and proofs are given specifically for the censored regression tobit model with a summary of analogous results for the case of random censoring and for truncated regression simulation results are also presented

application of bond moving renormalization group approach to fractal lattices we investigate the application of the migdal kadanoff bond moving renormalization group rg approach to fractal lattices we find the following two results first for inhomogeneous interaction lattice models bond moving involving inequivalent bonds is unsuitable because it violates the condition 0 is the perturbation potential resulting from moving the bonds second the condition 0 does not uniquely determine the way to move bonds different choices of bond moving yield different rg recursion relations and corresponding fixed points which makes the conclusions concerning the phase transition quite uncertain

application of the operational matrix of fractional order legendre functions for solving the time fractional convection diffusion equation the operational matrix of fractional order legendre functions method are considered the fractional order convection diffusion problem is solved the problem can be used extensively in science and engineering as in oil reservoir simulations problems which an initially discontinuous profile is propagated by diffusion and convection in this paper the application of the operational matrix of fractional order legendre functions flfs to solve the time fractional convection diffusion equation has been investigated fractional calculus has been applied to model the engineering and physical processes which are best described with other mathematical tools the time variable of the time fractional convection diffusion equation and its space variable have been approximated by flfs and shifted legendre polynomials respectively the fractional derivatives together with product matrices of flfs are employed to convert the solution of this problem to the solution of a system of algebraic equations

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

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

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

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

an inverse heat conduction problem with a nonlinear source term abstract in this paper the nonlinear problem of an inhomogeneous heat equation with linear boundary conditions will be considered the surface heat flux history of a heated conducting body will be identified the approach of the proposed method is to approximate the unknown function using linear polynomial pieces which are determined consecutively from the solution of the minimization problem on the basis of overspecified data some numerical examples will also be presented

quantile regression under random censoring censored regression models have received a great deal of attention in both the theoretical and applied econometric literature most of the existing estimation procedures for either cross sectional or panel data models are designed only for models with fixed censoring in this paper a new procedure for adapting these estimators designed for fixed censoring to models with random censoring is proposed this procedure is then applied to the clad and quantile estimators of powell j econom 25 1984 303 32 1986a 143 to obtain an estimator of the coefficients under a mild conditional quantile restriction on the error term that is applicable to samples exhibiting fixed or random censoring the resulting estimator is shown to have desirable asymptotic properties and performs well in a small scale simulation study

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

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

adaptive estimation in time series regression models this work develops adaptive estimators for a linear regression model with serially correlated errors we show that these results continue to hold when the order of the arma process characterizing the errors is unknown the finite sample results are promising indicating that substantial efficiency gains may be possible for samples as small as 50 observations we use these estimators to investigate the behavior of the forward foreign exchange market

semiparametric estimation of the type 3 tobit model abstract this paper considers estimation of the type 3 tobit model with only some weak restrictions imposed on the distribution of the error terms two least squares type estimation approaches are proposed under the condition that the error terms and regressors are independent consistent estimators for the asymptotic covariance matrices are proposed to facilitate large sample inference and a small monte carlo simulation is performed to investigate the finite sample behavior of the proposed estimators also the semiparametric efficiency bound is derived for the type 3 tobit model under the independence restriction

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

application jacobi spectral method for solving the time fractional differential equation abstract our aim from this article is to propose a spectral method for numerically solving a general form of time fractional differential equation with boundary conditions our method is based on the shifted jacobi polynomials and has four steps as follows at first the problem is converted to a new problem with homogeneous conditions step 1 then an integro differential equation is obtained which is equivalent with the new problem step 2 next all the known and unknown functions which are existing in the integro differential equation are approximated via the shifted jacobi polynomials step 3 utilizing operational matrices of shifted jacobi polynomials and collocation method a system of nonlinear algebraic equations is achieved which is solved by newton s iterative method step 4 in order to show and verify the accuracy and performance of the proposed method we examine our method on some illustrative examples

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

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

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

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

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

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

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

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

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

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

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

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

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

uniform laws of large numbers and stochastic lipschitz continuity this paper presents weak and strong uniform laws of large numbers for stochastically lipschitz continuous averages of functions of independent and dependent random variables it extends results from the literature by making use of the cover number properties of stochastically lipschitz continuous averages on the one hand and convergence speeds of averages of bounded functions on the other hand the resulting conditions for uniform laws of large number for stochastically lipschitz continuous averages of functions trade off heterogeneity and dependence against trending in the lipschitz function of the function class involved