bellman equations associated to the optimal feedback control of stochastic navier stokes equations in this paper we study infinite dimensional second order hamilton jaco bi bellman equations associated to the feedback synthesis of stochastic navier stokes equations forced by space time white noise uniqueness and existence of viscosity solutions are proven for these infinite dimensional partial differential equations c 2004 wiley periodicals inc

bandwidth selection for high dimensional covariance matrix estimation the banding estimator of bickel and levina and its tapering version of cai zhang and zhou are important high dimensional covariance estimators both estimators require a bandwidth parameter we propose a bandwidth selector for the banding estimator by minimizing an empirical estimate of the expected squared frobenius norms of the estimation error matrix the ratio consistency of the bandwidth selector is established we provide a lower bound for the coverage probability of the underlying bandwidth being contained in an interval around the bandwidth estimate extensions to the bandwidth selection for the tapering estimator and threshold level selection for the thresholding covariance estimator are made numerical simulations and a case study on sonar spectrum data are conducted to demonstrate the proposed approaches supplementary materials for this article are available online

a comparison principle for hamilton jacobi equations related to controlled gradient flows in infinite dimensions we develop new comparison principles for viscosity solutions of hamilton jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures our examples are optimal control of ginzburg landau and fokker planck equations they arise in limit considerations of externally forced non equilibrium statistical mechanics models or through the large deviation principle for interacting particle systems our approach is based on two key ingredients an appropriate choice of geometric structure defining the gradient flow and a free energy inequality resulting from such gradient flow structure the approach allows us to handle hamiltonians with singular state dependency in the nonlinear term as well as hamiltonians with a state space which does not satisfy the radon nikodym property in the case where the state space is a hilbert space the method simplifies existing theories by avoiding the perturbed optimization principle

partial estimation of covariance matrices a classical approach to accurately estimating the covariance matrix of a p variate normal distribution is to draw a sample of size n p and form a sample covariance matrix however many modern applications operate with much smaller sample sizes thus calling for estimation guarantees in the regime n ll p we show that a sample of size n o m log6p is sufficient to accurately estimate in operator norm an arbitrary symmetric part of consisting of m n nonzero entries per row this follows from a general result on estimating hadamard products m where m is an arbitrary symmetric matrix

cholesky decompositions and estimation of a covariance matrix orthogonality of variance correlation parameters summary chen dunson 2003 have proposed a modified cholesky decomposition of the form e dll d for a covariance matrix where d is a diagonal matrix with entries proportional to the square roots of the diagonal entries of e and l is a unit lower triangular matrix solely determining its correlation matrix this total separation of variance and correlation is definitely a major advantage over the more traditional modified cholesky decomposition of the form ld2l pourahmadi 1999 we show that though the variance and correlation parameters of the former decomposition are separate they are not asymptotically orthogonal and that the estimation of the new parameters could be more demanding computationally we also provide statistical interpretation for the entries of l and d as certain moving average parameters and innovation variances and indicate how the existing likelihood procedures can be employed to estimate the new parameters

generation of conditional densities in nonlinear filtering for infinite dimensional systems consider a signal process that takes values in an infinite dimensional space and assume that this process is observed through a nonlinear diffusion process in the nonlinear filtering theory literature recursive expressions for such problems in the form of conditional expectations and consequently measure valued stochastic partial differential equations on the underlying infinite dimensional space are obtained in this work state estimation problems for such stochastic systems are considered and nonlinear filtering equations in the form of conditional densities are obtained this is achieved by first deriving the filtering equations in the form of conditional expectations for a banach space valued state process whose dynamics depends on a finite dimensional brownian motion next a recursion for the conditional density which we assume to exist with respect to a sigma finite measure is obtained for such an analysis we obtain the adjoint operator for the generator of the state process which requires to use the theory of integration by parts in the infinite dimensional domains finally we discuss some potential application of the derived filtering equations in a portfolio optimization problem for portfolios which consist of bonds whose price curve is interpreted as a stochastic partial differential equation taking values in a banach space applications in mean field games are also demonstrated

parsimonious covariance matrix estimation for longitudinal data this article proposes a data driven method to identify parsimony in the covariance matrix of longitudinal data and to exploit any such parsimony to produce a statistically efficient estimator of the covariance matrix the approach parameterizes the covariance matrix through the cholesky decomposition of its inverse for longitudinal data this is a one step ahead predictive representation and the cholesky factor is likely to have off diagonal elements that are zero or close to zero a hierarchical bayesian model is used to identify any such zeros in the cholesky factor similar to approaches that have been successful in bayesian variable selection the model is estimated using a markov chain monte carlo sampling scheme that is computationally efficient and can be applied to covariance matrices of high dimension it is demonstrated through simulations that the proposed method compares favorably in terms of statistical efficiency with a highly regarded competing approach the estimator is applied to three

on the existence of optimal relaxed controls of stochastic partial differential equations this paper is concerned with control problem of systems governed by stochastic partial differential equations the drift and diffusion terms of which are second and first order differential operators respectively the existence of an optimal relaxed control is studied in both cases where the systems are degenerate and nondegenerate it is shown that the higher regularity conditions on the initial state as required in the existing results can be dispensed with if the wiener process is one dimensional some special cases of multidimensional wiener process are also discussed which in particular leads to an improvement of a recent result of bensoussan and nisio the method is based on an analysis of the group generated by the first order differential operator as an application an existence theorem of the optimal relaxed control is proved for partially observed diffusions with correlation between the controlled states and the observation noises

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

multivariate generalized gaussian distribution convexity and graphical models we consider covariance estimation in the multivariate generalized gaussian distribution mggd and elliptically symmetric es distribution the maximum likelihood optimization associated with this problem is non convex yet it has been proved that its global solution can be often computed via simple fixed point iterations our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions our second contribution is a generalized framework for structured covariance estimation under sparsity constraints we show that the optimizations can be formulated as convex minimization as long the mggd shape parameter is larger than half and the sparsity pattern is chordal these include for example maximum likelihood estimation of banded inverse covariances in multivariate laplace distributions which are associated with time varying autoregressive processes

viscosity solutions of hamilton jacobi equations in infinite dimensions iv hamiltonians with unbounded linear terms in this work we show how to interpret s and e in the viscosity sense when the solutions may be expected to be weakly sequentially continuous in particular we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

mild solutions of semilinear elliptic equations in hilbert spaces abstract this paper extends the theory of regular solutions c 1 in a suitable sense for a class of semilinear elliptic equations in hilbert spaces the notion of regularity is based on the concept of g derivative which is introduced and discussed a result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property that is it maps continuous functions into g differentiable ones the validity of this smoothing assumption is fully discussed for the case of the ornstein uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature it is shown that the results apply to hamilton jacobi bellman hjb equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that in particular they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

a characterization of e optimal controls for stochastic systems with partial observations via the unnormalized conditional density abstract the paper considers optimal control for continuous time stochastic systems with noisy measurements of the state it is known that the optimum coat for this problem may not be achieved within the class of strict sense controls however allowing for tolerances of size e a characterization is given for e optimal controls in terms of a semigroup formulation involving the unnormalized conditional density of the distribution of the state given the observations

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

a duality analysis on stochastic partial differential equations abstract the duality equations of stochastic partial differential equations are solved in the sobolev space h m w 2 m r d and the h m norm estimates of the solutions are obtained as an application the h m norm estimates with negative m for the solutions of stochastic partial differential equations are derived

viscosity solutions of dynamic programming equations for the optimal control of the two dimensional navier stokes equations in this paper we study the infinite dimensional hamilton jacobi equation associated with the optimal feedback control of viscous hydrodynamics we resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution

viscosity solutions for a class of hamilton jacobi equations in hilbert spaces we study hamilton jacobi equations with an unbounded term in hilbert spaces we introduce a new variant of the notion of viscosity solution for a class of hamilton jacobi equations and obtain comparison and existence results for viscosity solutions we also examine the usefulness of the notion of viscosity solution in optimal control

optimal relaxed controls for infinite dimensional stochastic systems of zakai type in this paper we present some new results on partially observed control problems for infinite dimensional stochastic systems in hilbert space using a fundamental result of da prato and zabczyk on an infinite dimensional kolmogorov operator we prove the existence of optimal relaxed controls for an infinite dimensional zakai equation following a semigroup approach and the theory of measurable selections this result is also extended to differential inclusion we also present some necessary conditions of optimality

conditional scores and optimal scores for generalized linear measurement error models abstract this paper studies estimation of the parameters of generalized linear models in canonical form when the explanatory vector is measured with independent normal error for the functional case i e when the explanatory vectors are fixed constants unbiased score functions are obtained by conditioning on certain sufficient statistics this work generalizes results obtained for logistic regression in the case that the explanatory vectors are independent and identically distributed with unknown distribtuion efficient score functions are obtained using the theory developed in begun et al 1983 keywords conditional score function efficient score function functional model generalized linear model measurement error structural model

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

stochastic uncertain systems subject to relative entropy constraints induced norms and monotonicity properties of minimax games entropy and relative entropy are fundamental concepts on which information theory is founded on and in general telecommunication systems design on the other hand dissipation inequalities minimax strategies and induced norms are the basic concepts on which robustness of uncertain control and estimation of systems are founded on in this paper the precise relation between these notions is investigated in particular it will be shown that the higher the dissipation the higher the entropy of the system which has implications in computing the induced norm associated with robustness these connections are obtained by considering stochastic optimal uncertain control systems in which uncertainty is described by a relative entropy constraint between the nominal and uncertain measures while the pay off is a linear functional of the uncertain measure this is a minimax game in which the controller measure seeks to minimize the pay off while the disturbance measure aims at maximizing the pay off salient properties of the minimax solution are derived including a characterization of the optimal sensitivity reduction computation of the induced norm monotonicity properties of minimax solution and relations between dissipation and relative entropy of the system the theory is developed in an abstract setting and then applied to nonlinear partially observable continuous time uncertain controlled systems in which the nominal and uncertain systems are described by conditional distributions in addition existence of the optimal control policy among the class of policies known as wide sense control laws is shown and an explicit formulae for the worst case conditional measure is derived the results are applied to linear quadratic gaussian problems

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

a multiple testing approach to the regularisation of large sample correlation matrices abstract this paper proposes a regularisation method for the estimation of large covariance matrices that uses insights from the multiple testing m t literature the approach tests the statistical significance of individual pair wise correlations and sets to zero those elements that are not statistically significant taking account of the multiple testing nature of the problem the effective p values of the tests are set as a decreasing function of n the cross section dimension the rate of which is governed by the nature of dependence of the underlying observations and the relative expansion rates of n and t the time dimension in this respect the method specifies the appropriate thresholding parameter to be used under gaussian and non gaussian settings the m t estimator of the sample correlation matrix is shown to be consistent in the spectral and frobenius norms and in terms of support recovery so long as the true covariance matrix is sparse the performance of the proposed m t estimator is compared to a number of other estimators in the literature using monte carlo experiments it is shown that the m t estimator performs well and tends to outperform the other estimators particularly when n is larger than t

a semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error summary a common objective in longitudinal studies is to characterise the relationship between a failure time process and time independent and time dependent covariates timedependent covariates are generally available as longitudinal data collected periodically during the course of the study we assume that these data follow a linear mixed effects model with normal measurement error and that the hazard of failure depends both on the underlying random effects describing the covariate process and other time independent covariates through a proportional hazards relationship a routine assumption is that the random effects are normally distributed however this need not hold in practice within this framework we develop a simple method for estimating the proportional hazards model parameters that requires no assumption on the distribution of the random effects large sample properties are discussed and finite sample performance is assessed and compared to competing methods via simulation

an admissible systems approach to separation in partially observed stochastic control problems abstract stochastic optimal control of systems subject to noisy measurements of the states is considered the formulation is based on the concept of admissible systems it is shown that the costs achieved in the class of wide sense admissible systems and in the class of separated admissible systems are equal over a fixed discretization of the time interval it is also shown that strongly separated controls can be used to achieve e optimality

classes of nonlinear partially observable stochastic optimal control problems with explicit optimal control laws this paper introduces certain nonlinear partially observable stochastic optimal control problems which are equivalent to completely observable control problems with finite dimensional state space in some cases the optimal control laws are analogous to linear exponential quadratic gaussian and linear quadratic gaussian tracking problems the problems discussed allow nonlinearities to enter the unobservable dynamics as gradients of potential functions the methodology is based on explicit solutions of a modified duncan mortensen zakai equation

partially observed control of markov processes iv abstract a class of degenerate elliptic nonlinear second order equations is considered these equations arising from a related class of probabilistic variational problems are defined on the space of probability measures on rd the existence of c1 1 viscosity solutions is established as well as the existence uniqueness and feedback characterization of the minima of the variational problems

average cost dynamic programming equations for controlled markov chains with partial observations the value function for the average cost control of a class of partially observed markov chains is derived as the vanishing discount limit in a suitable sense of the value functions for the corresponding discounted cost problems the limiting procedure is justified by bounds derived using a simple coupling argument

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

on filtering equations in infinite dimensions abstract we consider a nonlinear filtering problem for a state processxin a hilbert spaceh given a finite dimensional observation processy we look for solutions of the corresponding filtering equations having densities with respect to a gaussian reference measure inh we derive the equations for the conditional densities and prove that they have solutions we can allow correlated noise in the state and observation equations provided additional assumptions hold

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

on a functional analysis approach to parabolic equations in infinite dimensions abstract we consider here parabolic equations with infinitely many variables by using only analytic tools we prove existence uniqueness and regularity of solutions

testing super diagonal structure in high dimensional covariance matrices abstract the covariance matrices are essential quantities in econometric and statistical applications including portfolio allocation asset pricing and factor analysis testing the entire covariance under high dimensionality endures large variability and causes a dilution of the signal to noise ratio and hence a reduction in the power we consider a more powerful test procedure that focuses on testing along the super diagonals of the high dimensional covariance matrix which can infer more accurately on the structure of the covariance we show that the test is powerful in detecting sparse signals and parametric structures in the covariance the properties of the test are demonstrated by theoretical analyses simulation and empirical studies

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

information states in stochastic control and filtering a lie algebraic theoretic approach the purpose of the paper is two fold i to introduce the sufficient statistic algebra which is responsible for propagating the sufficient statistics or information state in the optimal control of stochastic systems and ii to apply certain lie algebraic methods and gauge transformations widely used in nonlinear control theory and quantum physics to derive new results concerning finite dimensional controllers this enhances our understanding of the role played by the sufficient statistics the sufficient statistic algebra enables us to determine a priori whether there exist finite dimensional controllers it also enables us to classify all finite dimensional controllers relations to minimax dynamic games are delineated

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

nonlinear semigroups associated with optimal stopping of controlled diffusions under partial observation abstract we treat optimal stopping problems of controlled diffusions under partial observation two kinds of admissibility strict sense and wide sense are considered using the girsanov transformation the unnormalized conditional probability turns out the state process subject to the zakai equation and we can derive a nonlinear semigroup which presents the bellman principle in our problems in the diffusion case an optimal stopping time is given appealing to a snell envelope

nonlinear kolmogorov equations in infinite dimensional spaces the backward stochastic differential equations approach and applications to optimal control solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control

multitask quantile regression under the transnormal model abstractwe consider estimating multitask quantile regression under the transnormal model with focus on high dimensional setting we derive a surprisingly simple closed form solution through rank based covariance regularization in particular we propose the rank based l1 penalization with positive definite constraints for estimating sparse covariance matrices and the rank based banded cholesky decomposition regularization for estimating banded precision matrices by taking advantage of the alternating direction method of multipliers nearest correlation matrix projection is introduced that inherits sampling properties of the unprojected one our work combines strengths of quantile regression and rank based covariance regularization to simultaneously deal with nonlinearity and nonnormality for high dimensional regression furthermore the proposed method strikes a good balance between robustness and efficiency achieves the oracle like convergence rate and provides the provable prediction interval und

partial information state based optimization of partially observable markov decision processes and the separation principle we propose a partial information state based approach to the optimization of the long run average performance in a partially observable markov decision process pomdp in this approach the information history is summarized at least partially by a or a few statistic s not necessary sufficient called a partial information state and actions depend on the partial information state rather than system states we first propose the single policy based comparison principle under which we derive an hjb type of optimality equation and policy iteration for the optimal policy in the partial information state based policy space we then introduce the q sufficient statistics and show that if the partial information state is q sufficient then the optimal policy in the partial information state based policy space is optimal in the space of all feasible information state based policies we show that with some further conditions the well known separation principle holds the results are obtained by applying the direct comparison based approach initially developed for discrete event dynamic systems

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

a moving average cholesky factor model in covariance modelling for longitudinal data we propose new regression models for parameterizing covariance structures in longitudinal data analysis using a novel cholesky factor the entries in this decomposition have a moving average and log innovation interpretation and are modelled as linear functions of covariates we propose efficient maximum likelihood estimates for joint mean covariance analysis based on this decomposition and derive the asymptotic distributions of the coefficient estimates furthermore we study a local search algorithm computationally more efficient than traditional all subset selection based on bic for model selection and show its model selection consistency thus a conjecture of pan mackenzie 2003 is verified we demonstrate the finite sample performance of the method via analysis of data on cd4 trajectories and through simulations

open loop evasion strategies in a pursuit evasion problem in a reduced state space abstract a stochastic pursuit evasion optimal control problem in the vertical plane is considered the pursuer has only partial observation of the evader s state whereas the evader knowing that he is being pursued applies a given open loop evasion strategy by reducing the state space of the encounter a method is proposed for evaluating the effectiveness of the open loop evasion strategies

convex banding of the covariance matrix abstractwe introduce a new sparse estimator of the covariance matrix for high dimensional models in which the variables have a known ordering our estimator which is the solution to a convex optimization problem is equivalently expressed as an estimator that tapers the sample covariance matrix by a toeplitz sparsely banded data adaptive matrix as a result of this adaptivity the convex banding estimator enjoys theoretical optimality properties not attained by previous banding or tapered estimators in particular our convex banding estimator is minimax rate adaptive in frobenius and operator norms up to log factors over commonly studied classes of covariance matrices and over more general classes furthermore it correctly recovers the bandwidth when the true covariance is exactly banded our convex formulation admits a simple and efficient algorithm empirical studies demonstrate its practical effectiveness and illustrate that our exactly banded estimator works well even when the true covariance

partially observed control of markov processes iii let v denote the value function of a partially observed control problem if v is one differentiable in a certain direction b then optimal controls are characterized by a feedback involving the directional derivative b v it is also shown that v satisfies the corresponding bellman equation an infinite dimensional pde on the space of measures in the viscosity sense of crandall and lions

discrete time survival data with longitudinal covariates survival analysis has been conventionally performed on a continuous time scale in practice the survival time is often recorded or handled on a discrete scale when this is the case the discrete time survival analysis would provide analysis results more relevant to the actual data scale besides data on time dependent covariates in the survival analysis are usually collected through intermittent follow ups resulting in the missing and mismeasured covariate data in this work we propose the sufficient discrete hazard sdh approach to discrete time survival analysis with longitudinal covariates that are subject to missingness and mismeasurement the sdh method employs the conditional score idea available for dealing with mismeasured covariates and the penalized least squares for estimating the missing covariate value using the regression spline basis the sdh method is developed for the single event analysis with the logistic discrete hazard model and for the competing risks analysis with the multinomial logit model simulation results revel good finite sample performances of the proposed estimator and the associated asymptotic theory the proposed sdh method is applied to the scleroderma lung study data where the time to medication withdrawal and time to death were recorded discretely in months for illustration

efficient estimation of covariance selection models summary a bayesian method is proposed for estimating an inverse covariance matrix from gaussian data the method is based on a prior that allows the off diagonal elements of the inverse covariance matrix to be zero and in many applications results in a parsimonious parameterisation of the covariance matrix no assumption is made about the structure of the corresponding graphical model so the method applies to both nondecomposable and decomposable graphs all the parameters are estimated by model averaging using an efficient metropolis hastings sampling scheme a simulation study demonstrates that the method produces statistically efficient estimators of the covariance matrix when the inverse covariance matrix is sparse the methodology is illustrated by applying it to three examples that are high dimensional relative to the sample size

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

whittle index for partially observed binary markov decision processes we consider the problem of dynamically scheduling m out of n binary markov chains when only noisy observations of state are available with ergodic equivalently long run average reward by passing on to the equivalent problem of controlling the conditional distribution of state given observations and controls it is cast as a restless bandit problem and its whittle indexability is established

two armed restless bandits with imperfect information stochastic control and indexability we present a two armed bandit model of decision making under uncertainty where the expected return to investing in the risky arm increases when choosing that arm and decreases when choosing the safe arm these dynamics are natural in applications such as human capital development job search and occupational choice using new insights from stochastic control along with a monotonicity condition on the payoff dynamics we show that optimal strategies in our model are stopping rules that can be characterized by an index which formally coincides with gittins index our result implies the indexability of a new class of restless bandit models

second order hamilton jacobi equations in hilbert spaces and stochastic boundary control the paper is concerned with fully nonlinear second order hamilton jacobi bellman isaacs equations of elliptic type in separable hilbert spaces which have unbounded first and second order terms the viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved a stochastic optimal control problem driven by a parabolic stochastic pde with control of dirichlet type on the boundary is considered it is proved that the value function of this problem is the unique viscosity solution of the associated hamilton jacobi bellman equation

a profile likelihood approach for longitudinal data analysis summary inappropriate choice of working correlation structure in generalized estimating equations gee could lead to inefficient parameter estimation while impractical normality assumption in likelihood approach would limit its applicability in longitudinal data analysis in this article we propose a profile likelihood method for estimating parameters in longitudinal data analysis via maximizing the estimated likelihood the proposed method yields consistent and efficient estimates without specifications of the working correlation structure nor the underlying error distribution both theoretical and simulation results confirm the satisfactory performance of the proposed method we illustrate our methodology with a diastolic blood pressure data set

adaptive boundary concentration control using zakai equation a mean variance control problem is formulated with respect to a partially observed nonlinear system that includes unknown constant parameters a physical prototype of the system is the cathode surface reaction in an electrolysis cell where the controller aim is to keep the boundary concentration of species in the near vicinity of the cathode surface low but not zero the boundary concentration is a diffusion controlled process observed through the measured current density and in practice controlled through the applied voltage the former incomplete data control problem is converted to complete data to the so called separated control problem whose solution is given by the infinite dimensional zakai equation in this article the separated control problem is solved numerically using pathwise integration of the zakai equation this article demonstrates precise tracking of the target trajectory with a rapid convergence of estimates to unknown parameters which take place simultaneously with control

optimal control for partially observed diffusions stochastic control problems are considered in which a state process x t and an observation process y t are governed by ito sense stochastic differential equations driven by independent brownian motions the control u t enters linearly in the dynamics of x t a separated control problem is introduced in which the state at any time t is a measure lambda t representing an unnormalized conditional distribution for x t given y s u s for s leqq t the method depends on introducing a pathwise version of lambda t which depends continuously on observation and control trajectories y u existence of an optimal control is obtained in a suitable class larger than the usual class of controls admissible in the strict sense that u t is measurable on the sigma algebra mathcal f t y generated by observations y s s leqq t the dynamics of lambda t are studied using a method of forward and backward partial differential equations under a suitable nondegeneracy condition the m

optimal controls for stochastic partial differential equations this paper deals with control problems for a state process governed by controlled linear stochastic partial differential equations whose drift and diffusion coefficients are the second order elliptic and the first order differential operators respectively a relaxed system is introduced as a generalization of admissible control and the continuous dependence of state process on a relaxed system assuming some regularity conditions is proved appealing to the usual compactification method this continuity result derives the existence of an optimal relaxed system and under convexity condition of coefficients an optimal relaxed system provides an optimal admissible control in a wider sense a relaxed control can be approximated by an admissible control which is brownian adapted and where bellman principle holds as an application stochastic control of diffusions with partial observation where the state noise and the observation noise may not be independent is discussed

an adaptive two sample test for high dimensional means several two sample tests for high dimensional data have been proposed recently but they are powerful only against certain limited alternative hypotheses in practice since the true alternative hypothesis is unknown it is unclear how to choose a powerful test we propose an adaptive test that maintains high power across a wide range of situations and study its asymptotic properties its finite sample performance is compared with existing tests we apply it and other tests to detect possible associations between bipolar disease and a large number of single nucleotide polymorphisms on each chromosome based on a genome wide association study dataset numerical studies demonstrate the superior performance and high power of the proposed test across a wide spectrum of applications

semiparametric marginal regression analysis for dependent competing risks under an assumed copula summary competing risks problems arise in many fields of science where two or more types of event may occur on a subject but only the event occurring first is observed together with its occurrence time and other events are censored the marginal and joint distributions of event times for competing risks cannot be identified from the observed data without assuming the relationship between events the commonly adopted independent censoring assumption may be easily violated an alternative is to assume that the joint distribution of event times follows a known copula which is an explicit function of the marginal distributions on the basis of the latter assumption we consider marginal regression analysis for dependent competing risks with the marginal regressions performed via semiparametric transformation models including the proportional hazards and proportional odds models we propose a non parametric maximum likelihood analysis which provides explicit expressions for the score functions and information matrix and facilitates convenient computations large and finite sample properties are studied we report an illustration with data from an acquired immune deficiency syndrome clinical trial where the censoring may be dependent the proposal can be readily used as a sensitivity analysis for assessing effects of potential dependent censoring and can incorporate external information on the association of competing risks

on the relation of zakai s and mortensen s equations the problem of optimal control for partially observable diffusion processes is studied by the dynamic programming in function space approach first proposed by r e mortensen the density of the conditional distribution of the unobservable signal given past and present observations which satisfies the zakai equation of nonlinear filtering is viewed as the new state of the system a verification result is established for the corresponding bellman hamilton jacobi equation known as mortensen s equation

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

a joint model for survival and longitudinal data measured with error the relationship between a longitudinal covariate and a failure time process can be assessed using the cox proportional hazards regression model we consider the problem of estimating the parameters in the cox model when the longitudinal covariate is measured infrequently and with measurement error we assume a repeated measures random effects model for the covariate process estimates of the parameters are obtained by maximizing the joint likelihood for the covariate process and the failure time process this approach uses the available information optimally because we use both the covariate and survival data simultaneously parameters are estimated using the expectation maximization algorithm we argue that such a method is superior to naive methods where one maximizes the partial likelihood of the cox model using the observed covariate values it also improves on two stage methods where in the first stage empirical bayes estimates of the covariate process are computed and then used as time dependent covariates in a second stage to find the parameters in the cox model that maximize the partial likelihood

sparse estimation of a covariance matrix we suggest a method for estimating a covariance matrix on the basis of a sample of vectors drawn from a multivariate normal distribution in particular we penalize the likelihood with a lasso penalty on the entries of the covariance matrix this penalty plays two important roles it reduces the effective number of parameters which is important even when the dimension of the vectors is smaller than the sample size since the number of parameters grows quadratically in the number of variables and it produces an estimate which is sparse in contrast to sparse inverse covariance estimation our method s close relative the sparsity attained here is in the covariance matrix itself rather than in the inverse matrix zeros in the covariance matrix correspond to marginal independencies thus our method performs model selection while providing a positive definite estimate of the covariance the proposed penalized maximum likelihood problem is not convex so we use a majorize minimize approach in which we iteratively solve convex approximations to the original nonconvex problem we discuss tuning parameter selection and demonstrate on a flow cytometry dataset how our method produces an interpretable graphical display of the relationship between variables we perform simulations that suggest that simple elementwise thresholding of the empirical covariance matrix is competitive with our method for identifying the sparsity structure additionally we show how our method can be used to solve a previously studied special case in which a desired sparsity pattern is prespecified

on the necessary conditions of optimal controls for stochastic partial differential equations this paper concerns optimal control of systems governed by stochastic partial differential equations in which drift and diffusion terms are second and first order differential operators respectively necessary conditions for an optimal control are derived for both nondegenerate and degenerate systems and all the coefficients appearing in the equations are allowed to depend on the control variables furthermore the results obtained in the paper can also be used to derive necessary conditions of optimality for partially observed diffusions with correlation between the signals and the observation noises

hypothesis testing for band size detection of high dimensional banded precision matrices many statistical analysis procedures require a good estimator for a high dimensional covariance matrix or its inverse the precision matrix when the precision matrix is banded the cholesky based method often yields a good estimator of the precision matrix one important aspect of this method is determination of the band size of the precision matrix in practice crossvalidation is commonly used however we show that crossvalidation not only is computationally intensive but can be very unstable in this paper we propose a new hypothesis testing procedure to determine the band size in high dimensions our proposed test statistic is shown to be asymptotically normal under the null hypothesis and its theoretical power is studied numerical examples demonstrate the effectiveness of our testing procedure

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

the role of information state and adjoint in relating nonlinear output feedback risk sensitive control and dynamic games this paper employs logarithmic transformations to establish relations between continuous time nonlinear partially observable risk sensitive control problems and analogous output feedback dynamic games the first logarithmic transformation is introduced to relate the stochastic information state to a deterministic information state the second logarithmic transformation is applied to the risk sensitive cost function using the laplace varadhan lemma in the small noise limit this cost function is shown to be logarithmically equivalent to the cost function of an analogous dynamic game

an approximation method for controlled discrete jump processes under partial observations the paper is concerned with control and filtering of a discrete jump markov process when only the total number of jumps is observed a partially observable control problem is considered and equivalence between this problem and the separated one is proved a technique to approximate the value function under some suitable assumptions is presented

hamilton jacobi bellman equations for the optimal control of the duncan mortensen zakai equation abstract we study a class of hamilton jacobi bellman hjb equations associated to stochastic optimal control of the duncan mortensen zakai equation the equations are investigated in weighted l 2 spaces we introduce an appropriate notion of weak viscosity solution of such equations and prove that the value function is the unique solution of the hjb equation we apply the results to stochastic optimal control problems with partial observation and correlated noise

second order hamilton jacobi equations in infinite dimensions some second order hamilton jacobi equations connected to stochastic optimal control problems for infinite dimensional systems driven by a white noise are studied a direct method to prove existence and uniqueness of mild solutions is developed then this solution is identified as the value function of the related stochastic control problem and a feedback formula for optimal controls is derived

a stochastic model for electrodeposition process with applications in filtering and boundary control electrodeposition is a complex partially observed mass transfer process driven by several surface reactions without exact model in this article the process uncertainties are described by a finite number of wiener processes in a stochastic model applied in the filtering and control problems these problems are solved as a boundary observation control problem based on a finite diffusion model with uncertainties in the domain interior and on the boundaries a mixed boundary problem is considered on an interval with the dirichlet data on one end bulk solution and neumann data on the other end cathode surface the concentration of oxidising species in the domain interior is unattainable for observations but the flux on the boundary electric current can be measured with a limited accuracy sensor error the total flux for the main and side reactions is controlled by the current density on the cathode surface the disturbing effect of the side reactions is modelled as a noise the concentration of speci

infinite dimensional hamilton jacobi equations and dirichlet boundary control problems of parabolic type the paper is concerned with an infinite dimensional hamilton jacobi equation this equation is related to boundary control problems of dirichlet type for semilinear parabolic systems the viscosity solution approach is adapted to the equation under consideration using the properties of fractional powers of generators of analytic semigroups an existence and uniqueness result for such problem is obtained

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions