on a problem concerning spacings let un n 1 2 be a sequence of independent uniformly distributed random variables on 0 1 and si n i 1 n be the spacings induced by the order statistics of u1 un 1 denote begin array l m l n rm max min s i j n rm 1 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x i mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x n l 1 rm 0 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x j l end array the exact distribution of m2 n is determined and based on analysis of something called fibonacci distribution the weak and almost sure convergence of the sequence m2 n are discussed furthermore the limiting distribution of ml n is determined for any fixed l and the equation p mathop lim limits n to infty rm nl n m l n n log n 1 1 is shown for some sequence ln

classification of probability laws by tail behavior abstract in this article i refine parzen s density quantile tail exponent classification of probability laws by tail behavior by subdividing his medium tailed class into medium short medium medium and medium long i give a physical interpretation of short medium and long tailed distributions in terms of the limiting size of the extreme spacings in a random sample from the distribution i show that this classification by the limiting size of extreme spacings fits nicely within the framework of my refinement of parzen s density quantile classification

the consistency of automatic kernel density estimates we consider the parzen rosenblatt kernel density estimate on rd with data dependent smoothing factor sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities f to be estimated when the smoothing factor is a function only of the sample size n it is shown that these conditions are also necessary a generalization of results by deheuvels the consistency of various automatic kernel density estimates is a simple consequence of these theorems

some criticisms of stochastic shrinkage and ridge regression with counterexamples the existence theorems for reduced mean squared error ridge regression and shrunken estimators are commonly used to justify the use of completely stochastic versions of these same estimators but there are no guarantees for improvement over the original unbiased estimators in particular this is demonstrated for a case where all possible values of the stochastic factor are known to satisfy the relevant inequality of the corresponding existence theorem examples of estimators with guaranteed improvement are given

dti segmentation via the combined analysis of connectivity maps and tensor distances we describe a novel approach to extract the neural tracts of interest from a diffusion tensor image dti compared to standard streamline tractography existing probabilistic methods are able to capture fiber paths that deviate from the main tensor diffusion directions at the same time tensor clustering methods are able to more precisely delimit the border of the bundle to the best of our knowledge we propose the first algorithm which combines the advantages supplied by probabilistic and tensor clustering approaches the algorithm includes a post processing step to limit partial volume related segmentation errors we extensively test the accuracy of our algorithm on different configurations of a dti software phantom for which we systematically vary the image noise the number of gradients the geometry of the fiber paths and the angle between adjacent and crossing fiber bundles the reproducibility of the algorithm is supported by the segmentation of the corticospinal tract of nine patients additional segmentations of the corticospinal tract the arcuate fasciculus and the optic radiations are in accordance with anatomical knowledge the required user interaction is comparable to that of streamline tractography which allows for an uncomplicated integration of the algorithm into the clinical routine

bayesian sequential parameter estimation by cognitive radar with multiantenna arrays in this paper we consider the problem of bayesian sequential parameter estimation of extended targets for cognitive radar with multi antenna arrays using adaptive waveforms the target is modeled as a complex gaussian random process using iterative waveform transmission the cognitive radar estimates the target s characteristic parameters and updates its probabilistic model based on new measurements the adaptive waveform is designed by minimizing the conditional entropy from the posterior density of the model parameters we analyze the performance of the developed bayesian sequential estimation algorithm and derive expressions for the signal to noise ratio gain the asymp totical posterior cramer rao bound and the mutual information gain the analysis and numerical simulations demonstrate that the adaptive sequential bayesian estimator yields accelerated convergence of the estimate towards its true value and a smaller estimation error compared with the conventional bayesian esti mator that uses fixed waveform transmission under gaussian or non gaussian noise

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

on robust analysis of a normal location parameter pericchi and smith considered a normal location parameter problem with double exponential and student t prior distributions these two prior distributions both belong to the class of scale mixtures of normal distributions and are useful in providing a robust analysis of the normal location parameter problem in this paper we extend the analysis to other scale mixtures of normal distributions such as the exponential power and the symmetric stable distributions

cross validation in density estimation considerable interest has been shown recently in the use of cross validatory procedures much of this attention is due to the efforts of stone 1974a b 1977a b who has provided a general discussion of the method of cross validatory choice dawid 1974 suggested that in certain circumstances cross validation might lead to inconsistency and some of these fears were confirmed by stone 1977b in the present paper we examine a popular cross validatory argument for selecting the smoothing parameter or window size in the nonparametric estimation of a density we derive results describing the order of magnitude of the cross validatory window and show that the resulting estimator will perform suboptimally our results are confirmed by monte carlo trials alternative techniques such as that proposed by silverman 1978a seem preferable habbema hermans van den broek 1974 suggested a method for discriminant analysis based on nonparametric density estimation they considered kernel type estimates and in the univariate case these usually have the form

the grenader estimator a nonasymptotic approach on examine certaines proprietes non asymtotiques de l estimateur de grenander d une densite decroissante f cet estimation est defini comme la pente du plus petit majorant concave de la fonction de repartition empirique on montre que son risque mesure avec perte l 1 est borne par une certaine fonctionnelle qui depend de f et du nombre d observations n le comportement asymptotique de la fonctionnelle quand n tend vers l infini est en accord avec les performances asymptotiques connues de l estimation de grenander

on overfitting generalization and randomly expanded training sets an algorithmic procedure is developed for the random expansion of a given training set to combat overfitting and improve the generalization ability of backpropagation trained multilayer perceptrons mlps the training set is k means clustered and locally most entropic colored gaussian joint input output probability density function estimates are formed per cluster the number of clusters is chosen such that the resulting overall colored gaussian mixture exhibits minimum differential entropy upon global cross validated shaping numerical studies on real data and synthetic data examples drawn from the literature illustrate and support these theoretical developments

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

when is conflict normal abstract when data conflicts with quantified prior beliefs seemingly small changes in the functional form of the prior and or likelihood can have a profound effect on posterior inferences this article provides results on asymptotic forms of the posterior when two information sources conflict in particular let x be from the likelihood p 2 x with an unknown location parameter with p 1 the prior on sufficient conditions on p 1 and p 2 are provided to ensure that as x the posterior is asymptotically normal the conditions cover all combinations in which p 1 and p 2 are proportional to exp x i i with i 1 conditions are also provided that ensure that as some of the data become extreme the posterior variance goes to 0 converges to a constant or diverges to infinity other asymptotic behavior involving conflicting information sources is also discussed

asymptotically optimal cells for a historgram on examine les proprietes de l histogramme lorsque les cellules sont arbitraires on obtient une expression asymptotique pour l infimum de l erreur quadratique moyenne integree mise sur toutes les partitions

bayes error estimation using parzen and k nn procedures the use of k nearest neighbor k nn and parzen density estimates to obtain estimates of the bayes error is investigated under limited design set conditions by drawing analogies between the k nn and parzen procedures new procedures are suggested and experimental results are given which indicate that these procedures yield a significant improvement over the conventional k nn and parzen procedures we show that by varying the decision threshold many of the biases associated with the k nn or parzen density estimates may be compensated and successful error estimation may be performed in spite of these biases experimental results are given which demonstrate the effect of kernel size and shape parzen the size of k k nn and the number of samples in the design set

a note on the superadditive and the subadditive transformations of aggregation functions we expand the theoretical background of the recently introduced superadditive and subadditive transformations of aggregation functions necessary and sufficient conditions ensuring that a transformation of a proper aggregation function is again proper are deeply studied and exemplified relationships between these transformations are also studied

estimation of unimodal densities without smoothness assumptions the grenander estimator of a decreasing density which is defined as the derivative of the concave envelope of the empirical c d f is known to be a very good estimator of an unknown decreasing density on the half line mathbb r when this density is not assumed to be smooth it is indeed the maximum likelihood estimator and one can get precise upper bounds for its risk when the loss is measured by the mathbb l 1 distance between densities moreover if one restricts oneself to the compact subsets of decreasing densities bounded by h with support on 0 l the risk of this estimator is within a fixed factor of the minimax risk the same is true if one deals with the maximum likelihood estimator for unimodal densities with known mode when the mode is unknown the maximum likelihood estimator does not exist any more we shall provide a general purpose estimator together with a computational algorithm for estimating nonsmooth unimodal densities its risk is the same as the risk of the grenander estimator based on the knowledge of the true mode plus some lower order term it can also cope with small departures from unimodality

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

computing kernel smoothed conditional quantiles from many observations abstract quantiles of a variable y conditional on another variable x when plotted against x can be a useful descriptive tool these plots give a quick impression of the functional form of the relation between x and the location spread and shape of the conditional distribution of y if several y are observed for each x then sample quantiles could be calculated for each x the resulting quantile plot may be quite noisy however and smoothing across x may be desired this article presents an algorithm that calculates kernel smoothed conditional quantiles with a cross validation choice of bandwidth for x it is computationally feasible for large data sets when x assumes a small number of values since it requires only one pass through the full data set the cross validation does not require a pass through the data because of simplifications arising from the l 1 loss function being based on absolute values the technique is illustrated by plotting the conditional quantiles of the net wealth of a sample of

mean squared error properties of generalized ridge estimators abstract mean squared errors are derived and examined for a number of generalized ridge gr regression estimators and the estimators are compared in a number of specific regression models the potential of gr estimators is discussed

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

recursive filtering with non gaussian noises the kalman filter is an optimal recursive filter although its optimality can only be claimed under the gaussian noise environment in this paper we consider the problem of recursive filtering with non gaussian noises one of the most promising schemes which was proposed by masreliez 1972 1975 uses the nonlinear score function as the correction term in the state estimate unfortunately the score function cannot be easily implemented except for simple cases in this paper a new method for efficient evaluation of the score function is developed the method employs an adaptive normal expansion to expand the score function followed by truncation of the higher order terms consequently the score function can be approximated by a few central moments the normal expansion is made adaptive by using the concept of conjugate recentering and the saddle point method it is shown that the approximation is satisfactory and the method is simple and practically feasible experimental results are reported to demonstrate the effectiveness of the new algorithm

dynamic generalized linear models and bayesian forecasting abstract dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and posterior distributions for the exponential family parameters this leads to the calculation of closed standard form predictive distributions for forecasting and model criticism the structure of the models depends on the time evolution of underlying state variables and the feedback of observational information to these variables is achieved using linear bayesian prediction methods data analytic aspects of the models concerning scale parameters and outliers are discussed and some applications are provided dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and

superdecomposition integrals this study introduces and discusses a new class of integrals based on superdecompositions of integrated functions including an analysis of their relationship with decomposition integrals which were introduced recently by even and lehrer the proposed superdecomposition integrals have several properties that are similar or dual with respect to decomposition integrals but they also have some significant differences the convex integral is obtained by considering all possible superdecompositions with no constraints on the applied sets which can be treated as the greatest convex homogeneous functional that is bounded from above by the measure we consider the relationship with the universal integral of klement et al is also discussed finally some possible generalizations are outlined

on locally adaptive density estimation abstract theoretical and practical aspects of the sample point adaptive positive kernel density estimator are examined a closed form expression for the mean integrated squared error is obtained through the device of preprocessing the data by binning with this expression the exact behavior of the optimally adaptive smoothing parameter function is studied for the first time the approach differs from most earlier techniques in that bias of the adaptive estimator remains o h 2 and is not improved to the rate o h 4 a practical algorithm is constructed using a modification of least squares cross validation simulated and real examples are presented including comparisons with a fixed bandwidth estimator and a fully automatic version of abramson s adaptive estimator the results are very promising

monte carlo study of three data based nonparametric probability density estimators abstract although the theoretical properties of modern nonparametric probability density estimators have been studied for 25 years there remains the practical problem of how to specify the amount of bias or smoothing in a density estimate based on a random sample in this paper we review and evaluate three recently developed data based algorithms that completely specify a density estimate from a random sample using monte carlo techniques we compare the statistical accuracy of these algorithms as measured by the integrated mean squared error in addition we examine the sensitivity of these algorithms to outliers and estimate computer time requirements one conclusion we draw is that the statistical accuracy of these data based algorithms seems comparable to levels predicted by theoretical models

exact and approximate posterior moments for a normal location parameter the forms of first and second posterior moments for a normal location parameter are identified for a rather general class of prior distributions exact and approximate illustrations are given where the prior distribution is double exponential or student t respectively

robust recursive estimation in the presence of heavy tailed observation noise a first order approximation is derived for the conditional prior distribution of the state of a discrete time stochastic linear dynamic system in the presence of contaminated normal observation noise this distribution is then used to derive a first order approximation of the conditional mean minimum variance estimator if the observation noise distribution can be represented as a member of the e contaminated normal neighborhood then the conditional prior is also to first order an analogous perturbation from a normal distribution whose first two moments are given by the kalman filter moreover the perturbation is itself of a special form combining distributions whose parameters are given by banks of parallel kalman filters and optimal smoothers

posterior cumulant relationships in bayesian inference involving the exponential family abstract for bayesian inference in one parameter contexts where either the likelihood or the prior has an exponential family form relationships are derived for posterior moments and cumulants of functions of both the canonical and the expectation parameters the identities exhibited generalize the simple relationships well known in the conjugate analysis case applications of these results are indicated in the areas of bayesian robustness and approximation in particular results are obtained on the behavior of the posterior distribution for a large observation generalizing work of meeden and isaacson

robust sequential approximate bayesian estimation summary an approximation to the sequential updating of the distribution of location parameters of a linear time series model is developed for non normal observations the behaviour of the resulting non linear recursive filtering algorithm is examined and shown to have certain desirable properties for a variety of non normal error distributions illustrative examples are given and relationships with previous work on robustness and sequential estimation are mentioned we consider here the problem of sequential estimation of the location vector of a linear time series model termed the dynamic linear model by harrison and stevens 1976 the straightforward exact analysis obtained by assuming normal error and prior structure is generally lost when alternative error distributions are adopted and yet considerations of realism or robustness may strongly suggest such non normal assumptions the multi state model of harrison and stevens provides an approximate analysis based on a discrete variance mixture of normal distributions an approach which has been extensively investigated in the engineering literature under the name of gaussian sum approximations see for example alspach and sorenson 1971 our aim in this paper is to provide an approximate tractible recursive updating procedure for the location parameters which is applicable to a wide range of symmetric unimodal error distributions and is both more realistic and more elegant than the discrete gaussian sum approach in particular for heavy tailed distributions our procedures provide approximate bayesian methods for time series analysis which extend considerably the work of masreliez and martin 1977 and have close connections with classical robustness ideas such as m estimation and influence functions

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm

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

a non gaussian state space model and application to prediction of records on developpe une classe de modeles qui generalisent le travail de smith 1979 et bathes 1975 pour des donnees censurees

on the existence of aggregation functions with given super additive and sub additive transformations abstract in this note we study restrictions on the recently introduced super additive and sub additive transformations a a and a a of an aggregation function a we prove that if a has a slightly stronger property of being strictly directionally convex then a a and a is linear dually if a is strictly directionally concave then a a and a is linear this implies for example the existence of pairs of functions f g sub additive and super additive on 0 n respectively with zero value at the origin and satisfying relatively mild extra conditions for which there exists no aggregation function a on 0 n such that a f and a g

laws of the iterated logarithm for nonparametric density estimators we establish a law of the iterated logarithm for a triangular array of independent random variables and apply it to obtain laws for a large class of nonparametric density estimators we consider the case of rosenblatt parzen kernel estimators trigonometric series estimators and orthogonal polynomial estimators in detail and point out that our technique has wider application

superadditive and subadditive transformations of integrals and aggregation functions we propose the concepts of superadditive and of subadditive transformations of aggregation functions acting on non negative reals in particular of integrals with respect to monotone measures we discuss special properties of the proposed transforms and links between some distinguished integrals superadditive transformation of the choquet integral as well as of the shilkret integral is shown to coincide with the corresponding concave integral recently introduced by lehrer similarly the transformation of the sugeno integral is studied moreover subadditive transformation of distinguished integrals is also discussed

akaike s information criterion and kullback leibler loss for histogram density estimation we take a detailed look at akaike s information criterion aic and kullback leibler cross validation klcv in the problem of histogram density estimation two different definitions of number of unknown parameters inaic are considered a careful description is given of the influence of density tail properties on performance of both types ofaic and onklcv a number of practical conclusions emerge in particular we find thataic will often give problems when used with heavy tailed unbounded densities but can perform quite well with compactly supported densities in the latter case both types ofaic produce similar results and those results will sometimes be asymptotically equivalent to the ones obtained fromklcv however depending on the shape of the true density theklcv method can fail to balance bias and variance components of loss with the result thatklcv andaic may produce very different results

robustification of kalman filter models abstract kalman filter models based on the assumption of multivariate gaussian distributions are known to be nonrobust this means that when a large discrepancy arises between the prior distribution and the observed data the posterior distribution becomes an unrealistic compromise between the two in this article we discuss a rationale for how to robustify the kalman filter specifically we develop a model wherein the posterior distribution will revert to the prior when extreme outlying observations are encountered and we point out that this can be achieved by assuming a multivariate distribution with student t marginals to achieve fully robust results of the kind desired it becomes necessary to forsake an exact distribution theory approach and adopt an approximation method involving poly t distributions a recursive mechanism for implementing the multivariate t based kalman filter is described its properties are discussed and the procedure is illustrated by an example

kernel density estimation and intrinsic alignment for shape priors in level set segmentation in this paper we make two contributions to the field of level set based image segmentation firstly we propose shape dissimilarity measures on the space of level set functions which are analytically invariant under the action of certain transformation groups the invariance is obtained by an intrinsic registration of the evolving level set function in contrast to existing approaches to invariance in the level set framework this closed form solution removes the need to iteratively optimize explicit pose parameters the resulting shape gradient is more accurate in that it takes into account the effect of boundary variation on the object s pose secondly based on these invariant shape dissimilarity measures we propose a statistical shape prior which allows to accurately encode multiple fairly distinct training shapes this prior constitutes an extension of kernel density estimators to the level set domain in contrast to the commonly employed gaussian distribution such nonparametric density estimators are suited to model aribtrary distributions we demonstrate the advantages of this multi modal shape prior applied to the segmentation and tracking of a partially occluded walking person in a video sequence and on the segmentation of the left ventricle in cardiac ultrasound images we give quantitative results on segmentation accuracy and on the dependency of segmentation results on the number of training shapes