spatiospectral concentration on a sphere we pose and solve the analogue of slepian s time frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or alternatively of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications as well as in medical imaging computer science cosmology and numerical analysis the spherical slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a fredholm integral equation in the spatial domain the associated eigenvalues are a measure of the spatiospectral concentration when the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a sturm liouville operator this enables the eigenfunctions to be computed extremely accurately and efficiently even when their area bandwidth product or shannon number is large in the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth the spherical concentration problem reduces to its planar equivalent which exhibits self similarity when the shannon number is kept invariant

differential operators commuting with finite convolution integral operators some nonabelian examples slepian landau and pollak found that a certain finite convolution integral operator on the real line commutes with a much simpler second order differential operator this opens the way to a detailed analysis of the space of time and band limited functions which has found applications in several fields here we exhibit some exact analogues of these commutation results with the real line replaced by either a non abelian group or a symmetric space the sphere may be the most natural example for the applications

stationary random fields with linear regressions we analyze and identify stationary fields with linear regressions and quadratic conditional variances we give sufficient conditions to determine one dimensional distributions uniquely as normal and as certain compactly supported distributions our technique relies on orthogonal polynomials which under our assumptions turn out to be a version of the so called continuous q hermite polynomials

expansions of e xz y ϵx and their applications to the analysis of elliptically contoured measures abstract we present a general theorem concerning the relationship between the first two conditional moments of some random variables conditioned upon a ramdom vector and the distribution of the conditioning random vector we apply this result to study properties of marginal densities of elliptically contoured measures and their relation to conditional variances

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

multivariate classes of life distributions in reliability theory four classes of life distributions which have been useful in describing aging where systems are assumed to have independent univariate component lifetimes are 1 the increasing failure rate ifr class 2 the increasing failure rate average ifra class 3 the new better than used nbu class and 4 the new better than used in expectation nbue class these classes together with multivariate analogs of the ifr and ifra classes are reviewed and discussed an up to date survey of the literature on this topic is included new multivariate definitions of the nbu and nbue classes are introduced

on the projections of isotropic distributions the class of distributions on rm 1 c m oo which are the m dimensional marginal distributions of orthogonally invariant distributions on rm n is characterized this result is then used to provide a partial answer to the following question given a symmetric distribution on r and an integer n 2 under what conditions will there exist a random vector x e rn such that a x has the given distribution up to a positive scale factor for all a 0 a e rm

the turning bands method for simulation of random fields using line generation by a spectral method the turning bands method tbm for the simulation of multidimensional random fields is presented these fields commonly occur in the monte carlo simulation of hydrologic processes particularly groundwater flow and mass transport the general tbm equations for two and three dimensional fields are derived with particular emphasis on the more complicated two dimensional case for stationary two dimensional fields the unidimensional line process is generated by a simple spectral method a technique which can be generally applied to any two dimensional covariance function and which is easily extended to anisotropic and areal averaged processes theoretically and by example the tbm is shown to be ergodic even for a finite number of lines and it is demonstrated that it rapidly converges to the true statistics of the field guidelines are presented for the selection of model parameters which will be helpful in the design of simulation experiments the tbm is compared to other methods in terms of cost and accuracy demonstrating that the tbm is as accurate as and much less expensive than multidimensional spectral techniques and more accurate than the most expensive approaches which use matrix inversion such as the nearest neighbor approach the unidimensional spectral technique presented here permits for the first time the inexpensive and accurate tbm simulation of any proper two dimensional covariance function and should be of some help in the stochastic analysis of hydrologic processes

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

canonical likelihoods a new likelihood treatment of nuisance parameters summary a new approach to the likelihood analysis of models with nuisance parameters is proposed based on a singular value decomposition of the likelihood function and is applied to several two parameter models the invariance of the approach under reparameterization is demonstrated and the paper concludes with comments on multiparameter models

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

remarks on properties of probability distributions determined by conditional moments in this paper we consider some properties of rotation invariant distributions onrn which are determined by a form of conditional moment of order 0 in particular we prove that the gaussian distribution is determined uniquely by its conditional moments and we investigate the related question of finiteness of exponential moments the case of general 0 appears to be more difficult to analyze than the case 2 studied previously by other authors

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

multivariate new better than used distributions a general theory of multivariate new better than used nbu distributions is developed using the general theory some multivariate nbu conditions are introduced and studied some of the results apply as well to multivariate increasing hazard rate average ihra and exponential distributions proofs of some related well known implications are simplified

on the stahel donoho estimator and depth weighted means of multivariate data the depth of multivariate data can be used to construct weighted means as robust estimators of location the use of projection depth leads to the stahel donoho estimator as a special case in contrast to maximal depth estimators the depth weighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature we also confirm through a finite sample study that the stahel donoho estimator achieves a desirable balance between robustness and efficiency at gaussian models

a stochastic process approach to multivariate reliability systems notions based on conditional stochastic order the purpose of this paper is to introduce extensions of the well known notions irf dfr nbu and nwu in reliability theory these extensions are applicable when considering complex multicomponent systems where the components are not necessarily behaving independently and where by continuous observation of the system its past at any time t is known to some degree the definitions are in terms of conditional stochastic order of the system s or its components residual lifetimes given the field corresponding to observed past

on the linearity of regression a stochastic process x x t t t is called spherically generated if for each random vector x x t 1 ldots x t n there exist a random vector y y1 ym with a spherical radially symmetric distribution and a matrix a such that x is distributed as ay x is said to have the linear regression property if x0 x1 xn is a linear function of x1 x n whenever the xj s are elements of the linear span of x it is shown that providing the linear span of x has dimension larger than two then x has the linear regression property if and only if it is spherically generated the class of symmetric stable processes which are spherically generated is shown to coincide with the class of socalled sub gaussian processes characterizing those stable processes having the linear regression property

the failure and hazard processes in multivariate reliability systems the component failures in a complex multi component system are treated in the paper as a multivariate point process and the system failure as a univariate point process derived from this our main concern is to extend the well known concept of a hazard function to this general case it is suggested that e g the system failure hazard be defined as a stochastic process viz the compensator of the system failure counting process relative to the fields describing the system s past the connections between such a hazard process and the lifetime distribution classes of arjas arjas e 1981 a stochastic process approach to multivariate reliability systems notions based on conditional stochastic order math oper res 6 263 276 are also discussed

comments on application of the conditional population mixture model to image segmentation in the above correspondence1 a maximum likelihood method is proposed for estimating class memberships and underlying statistical parameters within the context of distribution mixtures in the present comment it is pointed out that biases are incurred in parameter estimation that the class memberships and parameters are conceptually different and therefore that the so called standard mixture likelihood is to be preferred also in the correspondence 1 akaike s information criterion aic is used to choose the number of classes in the mixture here a brief theoretical caveat is issued

sampling models which admit a given general exponential family as a conjugate family of priors let k k be a family of sampling distributions for the data x on a sample space x which is indexed by a parameter and let f be a family of priors on then f is said to be conjugate for k if it is closed under sampling that is if the posterior distributions of given the data x belong to f for almost all x in this paper we set up a framework for the study of what we term the dual problem for a given family of priors f a subfamily of a general exponential family find the class of sampling models k for which f is conjugate in particular we show that k must be a general exponential family dominated by some measure q on x b where b is the borel field on x it is the class of such measures q that we investigate in this paper we study its geometric features and general structure and apply the results to some familiar examples

some new representations in bivariate exchangeability consider an array x x ij i j n of random variables and let u u ij and v v ij be orthogonal transformations affecting only finitely many coordinates say that x is separately rotatable if uxv t mathop limits d x for arbitrary u and v and jointly rotatable if this holds with u v restricting u and v to the class of permutations we get instead the property of separate or joint exchangeability processes on 2 0 1 or 0 1 2 are said to be separately or jointly exchangeable if the arrays of increments over arbitrary square grids have these properties for some of the above cases explicit representations have recently been obtained independently by aldous and hoover the aim of the present paper is to continue the work of these authors by deriving some new representations and by solving the associated uniqueness and continuity problems

on the variation with respect to a parameter of zeros of bessel and q bessel functions abstract we apply what is essentially the hellmann feynman theorem in a discrete setting to derive representations for the derivatives with respect to a parameter of the positive zeros of a family of entire functions these representations prove that the zeros are monotone functions of the parameters involved this family includes the bessel functions jv x and their basic analog jv 2 x q as a by product we show that j v 1 2 v 1 increases with v when v 1 jv 1 being the smallest positive zero of jv x we also use the lommel polynomials to derive improved lower bounds for the smallest positive zero of jv x

representations convolution square roots and spherical vectors abstract we extend results on square integrable representations of a locally compact group to subrepresentations of a representations induced from a character of a closed subgroup we do so working in the framework of quotient representations of algebras

projection based depth functions and associated medians order n uniform consistency depth regions and contours induced from projection based depth functions are investigated structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained affine equivariant multivariate medians induced from projection based depth functions are probed the limiting distributions as well as the strong and order n consistency of the sample projection medians are established the finite sample performance of projection medians is compared with that of a leading depth induced median the tukey halfspace median induced from the tukey halfspace depth function it turns out that with appropriate choices of univariate location and scale estimators the projection medians have a very high finite sample breakdown point and relative efficiency much higher than those of the halfspace median based on the results obtained it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators respectively

stochastic neutral evolution equations on hilbert spaces with partially observed relaxed control and their necessary conditions of optimality abstract in this paper we develop the necessary conditions of optimality for partially observed control problems governed by stochastic neutral evolution equations on hilbert spaces driven by relaxed controls

finite sample breakdown points of projection based multivariate location and scatter statistics finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics the stahel donoho statistics and the maronna yohai statistics the definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data the finite sample breakdown points of the multivariate statistics though are not directly related to those of the corresponding univariate location and scale statistics a uniform finite sample breakdown point concept is needed the median and the median absolute deviation about the median m a d are one possible choice for the univariate location and scale statistics respectively for sparse data sets in high dimensions though it is not recommended that the m a d be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum a simple modification to the m a d however is shown to alleviate this problem for various reasons one may wish to consider univariate location and scale statistics other than the median and the m a d respectively a very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous m estimates of location and scale new results on their breakdown properties are given in this paper implicit formulas for the breakdown points of monotonic simultaneous m estimates of location and scale are known and they tend to imply rather low breakdown points for smooth choices of the defining weight functions it is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous m estimates of location and scale furthermore explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic m estimates

necessary and sufficient optimality conditions for relaxed and strict control problems we consider a stochastic control problem where the set of strict classical controls is not necessarily convex and the system is governed by a nonlinear stochastic differential equation in which the control enters both the drift and the diffusion coefficients by introducing a new approach we establish necessary as well as sufficient conditions of optimality for two models the first concerns the relaxed controls which are measure valued processes in which an optimal solution exists the second is a particular case of the first and relates to strict control problems these results are given in the form of global stochastic maximum principle by using only the first order expansion and the associated adjoint equation this improves all of the previous works on the subject

the behavior of the stahel donoho robust multivariate estimator abstract the stahel donoho estimators t v of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w r where w is a weight function and r is a measure of outlyingness obtained by considering all univariate projections of the data it has a high breakdown point for all dimensions and order n consistency the asymptotic bias of v for point mass contamination for suitable weight functions is compared with that of rousseeuw s minimum volume ellipsoid mve estimator a simulation shows that for a suitable w t and v exhibit high efficiency for both normal and cauchy distributions and are better than their competitors for normal data with point mass contamination the performances of the estimators for detecting outliers are compared for both a real and a synthetic data set

predictive likelihood inference with applications summary conditional and marginal predictive likelihood are defined and used in predicting un observed or missing data a distinction is made between predictive and estimative goals of a data analysis such methodology and distinctions are illustrated in the context of outlier theory and mixed model ano va the prediction of unobserved or missing data is considered from a likelihood perspective a general predictive likelihood is proposed for such purposes and contrasted with parametric likelihood the resulting maximum likelihood predictor is also compared with the limiting imputation from the em algorithm the new methodology is illustrated and contrasted with estimative likelihood analysis in the context of outlier theory and mixed model ano va predictions are proposed that are based on conditional and marginal predictive likelihoods for the unobserved data which are predictive adaptations of the methods developed by kalbfleisch and sprott 1970 1973 these authors were concerned with the elimination of nuisance parameters in order to make inference about another set of structural parameters prediction in the frame work of parametric distributions involves the elimination of all parameters in such a way that inference about the unobserved data is possible therefore in the predictive setting all para meters are nuisance parameters and the unobserved data now play a role similar to that of the structural parameters various predictive approaches to statistical inference have been advocated principally by geisser 1971 1975 stone 1974 aitchison and dunsmore 1975 and guttman 1970 a survey is provided by geisser 1980 and a brief overview in a parametric setting is presented in the next section following this a conditional likelihood approach is developed and shown to be closely related to the lauritzen 1974 hinkley 1979 likelihood method but specified in a manner allowing for a wider range of applicability a marginal predictive likelihood procedure is also specified which when applicable essentially agrees with the conditional likelihood procedure for the examples considered for a general account of likelihood based inference see fraser

influence function and maximum bias of projection depth based estimators location estimators induced from depth functions increasingly have been pursued and studied in the literature among them are those induced from projection depth functions these projection depth based estimators have favorable properties among their competitors in particular they possess the best possible finite sample breakdown point robustness however robustness of estimators cannot be revealed by the finite sample breakdown point alone the influence function gross error sensitivity maximum bias and contamination sensitivity are also important aspects of robustness in this article we study these other robustness aspects of two types of projection depth based estimators projection medians and projection depth weighted means the latter includes the stahel donoho estimator as a special case exact maximum bias the influence function and contamination and gross error sensitivity are derived and studied for both types of estimators sharp upper bounds for the maximum bias and the influence functions are established comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors

algebraic heun operator and band time limiting we introduce the algebraic heun operator associated to any bispectral pair of operators this operator can be presented as a generic bilinear combination of these bispectral operators we show that the resulting operators are natural generalizations of the ordinary heun operator this leads to a simple construction of the operators commuting with the projection operators in problems of band time limiting and it gives a way to adapt a construction first used by perline in a purely finite setup to general situations we also extend this algebraic construction to cover some purely finite cases which lie beyond this scheme in particular for the anti krawtchouk polynomials in the latter case the commuting operator is represented by a pentadiagonal matrix which can be constructed explicitly as a polynomial of degree 4 of the pair of bispectral operators

a new property of reproducing kernels for classical orthogonal polynomials abstract we exhibit a second order differential operator commuting with the reproducing kernel n 0 t n n h n each time that n is one of the classical orthogonal polynomials jacobi laguerre hermite and bessel this is the analog of a known property in the study of time and band limited signals

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

approximate predictive likelihood summary a predictive likelihood is given which approximates both bayes and maximum likeli hood predictive inference by expansion of a posterior likelihood this synthesizes and extends previous results and is widely applicable the approximation usually differs from exact bayes posterior predictive density by op n 2 and from exact predictive likelihood by op n 1 but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found the results are applied to the prediction various treatments of the likelihood function form the basis of parametric inference but until recently few non bayesian attempts have been made to define versions of it suitable for prediction however lauritzen 1974 hinkley 1979 and more recently butler 1986 have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic unfortunately they apply only when sufficiency provides a genuine reduction of the data moreover even if for a given problem an exact predictive likelihood exists the calculations needed to derive it can be onerous this paper provides approximations to hinkley lauritzen predictive likelihood accur ate to op n 1 in many cases which are applicable in situations where maximum likelihood estimation is regular they also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data the approximations are derived from a posterior predictive density and may if prior informa tion is available be regarded as bayesian procedures their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum and is fast and accurate using usual numerical maximization methods there is a close connexion with the approximations to posterior moments and marginal densities given by tierney kadane 1986 who suggest the same approxima tion as used here but give no detailed results for prediction section 2 of this paper shows how laplace s method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood section 3 discusses a connection with conditional inference essentially the same approximation

on a notion of data depth based on random simplices for a distribution f on mathbb r p and a point x in mathbb r p the simplical depth d x is introduced which is the probability that the point x is contained inside a random simplex whose vertices are p 1 independent observations from f mathematically and heuristically it is argued that d x indeed can be viewed as a measure of depth of the point x with respect to f an empirical version of d cdot gives rise to a natural ordering of the data points from the center outward the ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and l statistics this generalized sample median and l statistics are affine equivariant

spatial medians depth functions and multivariate jensen s inequality for any given partial order in a d dimensional euclidean space under mild regularity assumptions we show that the intersection of closed generalized intervals containing more than 1 2 of the probability mass is a non empty compact interval this property is shared with common intervals on real line where the intersection is the median set of the underlying probability distribution so obtained multivariate medians with respect to a partial order can be observed as special cases of centers of distribution in the sense of type d depth functions introduced by y zuo and r serfling em ann stat bf 28 2000 461 482 we show that the halfspace depth function can be realized via compact convex sets or for example closed balls in place of halfspaces and discuss structural properties of halfspace and related depth functions and their centers among other things we prove that in general the maximal guaranteed depth is frac 1 d 1 as an application of these results we provide a jensen s type inequality for functions of several variables with medians in place of expectations which is an extension of the previous work by m merkle em stat prob letters bf 71 2005 277 281

convolution roots of radial positive definite functions with compact support a classical theorem of boas kac and krein states that a characteristic function with x 0 for x t admits a representation of the form x u y u y x dy x r where the convolution root u l 2 r is complex valued with u x 0 for x 2 the result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type this paper examines the boas kac representation under additional constraints if is real valued and even can the convolution root u be chosen as a real valued and or even function a complete answer in terms of the zeros of the fourier transform of is obtained furthermore the analogous problem for radially symmetric functions defined on r d is solved perhaps surprisingly there are compactly supported radial positive definite functions that do not admit a convolution root with half support however under the additional assumption of nonnegativity radially symmetric convolution roots with half support exist further results in this paper include a characterization of extreme points pointwise and integral bounds turan s problem and a unified solution to a minimization problem for compactly supported positive definite functions specifically if f is a probability density on r d whose characteristic function vanishes outside the unit ball then x 2 f x dx 0 4j 2 d 2 2 where j v denotes the first positive zero of the bessel function j v and the estimate is sharp applications to spatial moving average processes geostatistical simulation crystallography optics and phase retrieval are noted in particular a real valued half support convolution root of the spherical correlation function in r 2 does not exist

estimating the stable index alpha in order to measure tail thickness a critique stable laws are often fit to outlier prone data and if the index alpha is estimated to be much less than two then the normal law is rejected in favor of an infinite variance stable law this paper derives the theoretical properties of such a procedure when the true distribution is stable the distribution of the m l e of alpha is non regular if alpha 2 when the true distribution is not stable the estimate of alpha is not a robust measure of the rate of decrease of the tail probabilities a more robust procedure is developed and a statistic for describing and comparing the tail shapes of arbitrary samples is proposed

some results on the denoising problem in the elliptically distributed context the purpose of this paper is to revisit the denoising problem of a multivariate elliptically symmetric random vector corrupted by a multivariate elliptically symmetric noise the investigation begins with a review of multivariate elliptically distributed random vectors and of their basic but important properties a focus is made on the gaussian scale mixtures a subclass of the elliptical distributions in the second part the denoising problem is revisited in this elliptical context following two classical directions the minimum mean square error and the maximum a posteriori estimation approaches although subject to some restrictions this investigation extends the two recent studies by alecu and selesnick the practical use of the proposed estimators and some possible special behaviors are then discussed through various illustrative examples

centralized versus decentralized optimization of distributed stochastic differential decision systems with different information structures part i a general theory decentralized optimization of distributed stochastic dynamical systems with two or more controls of the decision makers dms has been an active area of research for over half a century although such decentralized optimization problems are often formulated utilizing static team and person by person pbp optimality criteria the corresponding static team theory results have not been extended to dynamical systems in this first part of the two part paper we derive team and pbp optimality conditions for distributed stochastic differential systems when the controls of the dms generate actions based on different information structures the necessary conditions are given by a hamiltonian system described by coupled backward and forward stochastic differential equations sdes and a conditional hamiltonian conditioned on the information structures available to the controls of the dms the sufficient conditions state that pbp optimality implies team optimality if the hamiltonian is convex in the state and or actions spaces of the controls of the dms we show existence of relaxed team optimal strategies when the information structures are not affected by the controls of the dms throughout the paper we discuss similarities to analogous optimality conditions of centralized decision or control of stochastic systems and we note a connections to mean field stochastic optimal control problems

wiener s problem for positive definite functions we study the sharp constant w n d in wiener s inequality for positive definite functions begin aligned int mathbb t n f 2 dx le w n d d 1 int d f 2 dx quad d subset mathbb t n end aligned wiener proved that w 1 delta delta infty delta in 0 1 2 hlawka showed that w n d le 2 n where d is an origin symmetric convex body we sharpen hlawka s estimates for d being the ball delta b n and the cube delta i n in particular we prove that w n delta b n le 2 0 401 cdots o 1 n we also obtain a lower bound of w n d moreover for a cube d frac 1 q i n with q 3 4 ldots we obtain that w n d 2 n our proofs are based on the interrelation between wiener s problem and the problems of turan and delsarte

depth weighted scatter estimators general depth weighted scatter estimators are introduced and investigated for general depth functions we find out that these affine equivariant scatter estimators are fisher consistent and unbiased for a wide range of multivariate distributions and show that the sample scatter estimators are strong and n consistent and asymptotically normal and the influence functions of the estimators exist and are bounded in general we then concentrate on a specific case of the general depth weighted scatter estimators the projection depth weighted scatter estimators which include as a special case the well known stahel donoho scatter estimator whose limiting distribution has long been open until this paper large sample behavior including consistency and asymptotic normality and efficiency and finite sample behavior including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators are thoroughly investigated the influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined unlike typical high breakdown competitors the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a bounded influence function and a moderate maximum bias curve comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and consequently represent very favorable choices of affine equivariant multivariate scatter estimators

projection estimates of multivariate location in this paper we study the maximum asymptotic bias of the projection estimate for multivariate location based on univariate estimates of location and dispersion in particular we study the projection estimate that uses the median and median absolute deviation about the median mad as univariate location and dispersion estimates respectively this estimator may be considered a natural affine equivariant multivariate median for spherical distributions the maximum bias of this estimate depends only on the marginal distributions and not on the dimension and is approximately twice the maximum bias of the univariate median we also show that for multivariate normal distributions its maximum bias compares favorably with those of the donoho stahel minimum volume ellipsoid and minimum covariance determinant estimates in all these cases the maximum bias increases with the dimension p

multidimensional trimming based on projection depth as estimators of location parameters univariate trimmed means are well known for their robustness and efficiency they can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy tailed models this paper introduces multidimensional trimmed means based on projection depth induced regions robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point the influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators it is found that the projection depth trimmed means are highly robust locally as well as globally asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions the strong consistency asymptotic representation and limiting distribution of the depth trimmed means are obtained relative to the mean and other leading competitors the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated simulation studies confirm the validity of the asymptotic efficiency results at finite samples

interval estimation in discrimination the multivariate normal equal covariance case summary a general approach to discrimination problems is described which emphasizes the viewpoint of individual cases and expresses the uncertainty with which individual decisions are made it is applied to the particular case of two multivariate normal distributions with a common covariance matrix an affine transformation to which the key quantities are invariant simplifies discussion of the distribution of the estimated log odds ratio and suggests a practically useful plot which generalizes the usual display of discriminant scores and has several attractive features an illustration is given two approximate interval estimation techniques for the true log odds ratio are described and evaluated in a range of practical situations by a simulation study some further applications of our general approach are briefly indicated

random arrays and functionals with multivariate rotational symmetries our aim is to extend schoenberg s classical theorem to higher dimensions by establishing representations of arbitrary separately or jointly rotatable continuous linear random functionals in terms of multiple wiener ito integrals and their tensor products this leads to similar representations for separately or jointly rotatable arrays and for separately or jointly exchangeable or spreadable random sheets

stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and driven by relaxed controls in this paper we consider nonconvex control problems of stochastic differential equations driven by relaxed controls adapted in the weak star sense to a current of sigma algebras generated by observable processes we cover in a unified way both continuous diffusion and jump processes we present existence of optimal controls before we construct the necessary conditions of optimality unlike some papers in this area using only functional analysis we develop a stochastic hamiltonian system of equations on a rigorous basis using the semimartingale representation theory and the riesz representation theorem leading naturally to the existence of the adjoint process which satisfies a backward stochastic differential equation in other words our approach predicts the existence of the adjoint process as a natural consequence of riesz representation theory ensuring at the same time the weak star measurability this is unlike other papers where the adjoint process is introduced before its existence is prov

lower bounds for contamination bias globally minimax versus locally linear estimation we study how robust estimators can be in parametric families obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families this lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators and it is tight or nearly tight in a variety of other settings such as scale estimation discrete exponential families and multiple linear regression the minimum variation distance estimator has contamination bias within a dimension free factor of this bound a second lower bound applies to locally linear estimates and implies that such estimates cannot be bias minimax among all fisher consistent estimates in higher dimensions in linear regression this class of estimates includes the familiar m estimates gm estimates and s estimates in discrete exponential families yet another lower bound implies that the proportion of zeros estimate has minimax bias if the median of the distribution is zero a common situation in some fields this bound also implies that the information standardized sensitivity of every fisher consistent estimate of the poisson mean and of the binomial proportion is unbounded

can the first two conditional moments identify a mean square differentiable process following a general theorem concerning the relationship between the first two conditional moments of some random variables and the distribution of the conditioning random vector we derive differential equations which involve the conditional regression and covariance of the mean square derivative of the process wt and the finite dimensional densities of this process we use these equations to define a subclass of m s differentiable processes having smooth conditional moments and finite dimensional densities and having the property that the first two conditional moments identify a given process completely among these are processes having linear first moments we show that under some differentiability assumptions these processes have elliptically contoured distributions

the influence function and maximum bias of tukey s median tukey s median is among one of the earliest known high breakdown point multivariate location statistics aside from its breakdown point though little else appears to be known about its robustness properties in this paper we investigate other aspects of tukey s median and in particular we derive and study its influence function and its maximum contamination bias function when judged by these other robustness criteria tukey s median again proves to be highly robust

the relaxed stochastic maximum principle in singular optimal control of diffusions this paper studies optimal control of systems driven by stochastic differential equations where the control variable has two components the first being absolutely continuous and the second singular our main result is a stochastic maximum principle for relaxed controls where the first part of the control is a measure valued process to achieve this result we establish first order optimality necessary conditions for strict controls by using strong perturbation on the absolutely continuous component of the control and a convex perturbation on the singular one the proof of the main result is based on the strict maximum principle ekeland s variational principle and some stability properties of the trajectories and adjoint processes with respect to the control variable

nbu processes with general state space waiting times for many kinds of events can be represented as first passage times for appropriate stochastic processes in some applications the most natural stochastic process is multidimensional in other applications it may be more natural to consider a stochastic process which takes on values in a more general space in this paper we consider new better than used nbu stochastic processes on partially ordered separable banach spaces that is processes which have nbu first passage times to closed upper sets various equivalent conditions for a process to be nbu are derived some of the results can be used also to characterize increasing hazard rate average ihra processes examples of such processes include processes with shocks and recovery and processes with random repair times some applications in reliability theory and in other areas of probability are given

the stochastic maximum principle for linear convex optimal control with random coefficients this paper considers a stochastic control problem with linear dynamics convex cost criterion and convex state constraint in which the control enters both the drift and diffusion coefficients these coefficients are allowed to be random and no l p bounds are imposed on the control an explicit solution for the adjoint equation and a global stochastic maximum principle are obtained for this model this is evidently the first version of the stochastic maximum principle that covers the consumption investment problem the mathematical tools are those of stochastic calculus and convex analysis when it is assumed as in other versions of the stochastic maximum principle that the admissible controls are square integrable not only a necessary but also a sufficient condition for optimality is obtained it is then shown that this particular case of the general model may be applied to solve a variety of problems in stochastic control including the linear regulator predicted miss and benes problems

min max bias robust regression abstract this paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon contamination neighborhoods for the joint distribution of the response and carriers two classes of estimates are treated 1 m estimates with bounded function rho applied to the scaled residuals using a very general class of scale estimates and 2 bounded influence function type generalized m estimates estimates in the first class are obtained as the solution of a minimization problem while estimates in the second class are specified by an estimating equation the first class of m estimates is sufficiently general to include both huber proposal 2 simultaneous estimates of regression coefficients and residuals scale and rousseeuw yohai s estimates of regression it is shown than an s estimate based on a jump function type rho solves the min max bias problem for the class of m estimates with very general scale this estimate is obtained by the minimization of the alpha quantile of the squared residuals where alpha epsilon depends on the fraction of contamination epsilon when epsilon approaches limit of 5 alpha epsilon approaches limit of 5 and the min max estimator approaches the least median of squared residuals estimator introduced by rousseeuw for the bounded influence class of gm estimates it is shown the a sign type nonlinearity yields the min max estimate this estimate coincides with the minimum gross error sensitivity gm estimate

the darboux process and time and band limiting for matrix orthogonal polynomials abstract we extend to a situation involving matrix valued orthogonal polynomials a scalar result that originates in work of claude shannon in laying the mathematical foundations of information theory and a remarkable series of papers by d slepian h landau and h pollak at bell labs in the 1960 s we show that in this case an algebraic miracle that plays a very important role in the classical case survives an application of the so called darboux process in the matrix valued context

a stochastic maximum principle for a markov regime switching jump diffusion model and its application to finance this paper develops a sufficient stochastic maximum principle for a stochastic optimal control problem where the state process is governed by a continuous time markov regime switching jump diffusion model we also establish the relationship between the stochastic maximum principle and the dynamic programming principle in a markovian case applications of the stochastic maximum principle to the mean variance portfolio selection problem are discussed

approximate tolerance intervals based on maximum likelihood estimates abstract let x 1 xn be independent random variables with distributions depending on a possibly multidimensional let y be an unobserved continuously distributed random variable whose distribution depends on a tolerance interval for y is desired satisfying p y e i x 1 xn a naive interval would estimate from the x s and construct the interval assuming that the estimate is exactly correct this article assumes standard regularity conditions and uses taylor approximations to construct correction terms of order 1 n the resulting interval is longer than the naive interval because it takes into account the uncertainty in the estimate of two examples one simple and one complex illustrate the method

stochastic initial boundary value problems subject to distributed and boundary noise and their optimal control abstract in this paper we consider a class of stochastic evolution equations arising from initial boundary value problems with both boundary and distributed noise we prove existence and regularity of mild solutions then we consider a controlled version of the model and prove the existence of optimal controls and develop the necessary conditions of optimality for partially observed problems using relaxed controls

elliptically contoured random variables and their application to the extension of the kalman filter abstract we summarize some properties of elliptically contoured e c random variables and indicate their possible application in engineering practice in particular after presenting some new results we apply them to modify the kalman filter the modified version is robust against the extension of the noises model from normal to elliptically contoured

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

a class of multivariate new better than used distributions a class of multivariate distributions with new better than used nbu marginals is introduced a number of necessary and sufficient conditions for a distribution to be a member in the class are given closure results which are useful for the identification or construction of members in the class are derived in particular simple proofs of some well known preservation properties of the class of the univariate nbu distributions are obtained two examples of replacement models that give rise to multivariate nbu distributions in the class are discussed

breakdown properties of location estimates based on halfspace depth and projected outlyingness we describe multivariate generalizations of the median trimmed mean and w estimates the estimates are based on a geometric construction related to projection pursuit they are both affine equivariant coordinate free and have high breakdown point the generalization of the median has a breakdown point of at least 1 d 1 in dimension d and the breakdown point can be as high as 1 3 under symmetry in contrast various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than 1 d 1 in dimension d

approximate inference in state space models with heavy tailed noise state space models have been successfully applied across a wide range of problems ranging from system control to target tracking and autonomous navigation their ubiquity stems from their modeling flexibility as well as the development of a battery of powerful algorithms for estimating the state variables for multivariate models the gaussian noise assumption is predominant due its convenient computational properties in some cases anyhow this assumption breaks down and no longer holds we propose a novel approach to extending the applicability of this class of models to a wider range of noise distributions without losing the computational advantages of the associated algorithms the estimation methods we develop parallel the kalman filter and thus are readily implemented and inherit the same order of complexity we derive all of the equations and algorithms from first principles in order to validate the performance of our approach we present specific instances of non gaussian state space models and test their performance on experiments with synthetic and real data

approximation of the laplace and stokes operators with dirichlet boundary conditions through volume penalization a spectral viewpoint we report the results of a study on the spectral properties of laplace and stokes operators modified with a volume penalization term designed to approximate dirichlet conditions in the limit when a penalization parameter eta tends to zero the eigenvalues and eigenfunctions are determined either analytically or numerically as functions of eta both in the continuous case and after applying fourier or finite difference discretization schemes for fixed eta we find that only the part of the spectrum corresponding to eigenvalues lambda lesssim eta 1 1 approaches dirichlet boundary conditions while the remainder of the spectrum is made of uncontrolled spurious wall modes the penalization error for the controlled eigenfunctions is estimated as a function of eta and lambda surprisingly in the stokes case we show that the eigenfunctions approximately satisfy with a precision o eta o navier slip boundary conditions with slip length equal to sqrt eta moreover for a given discretization we show that there exists a value of eta corresponding to a balance between penalization and discretization errors below which no further gain in precision is achieved these results shed light on the behavior of volume penalization schemes when solving the navier stokes equations outline the limitations of the method and give indications on how to choose the penalization parameter in practical cases