some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

phase transition with the line tension effect we make the connection between the geometric model for capillarity with line tension and the cahn hilliard model of two phase fluids to this aim we consider the energies f eps u eps int o du 2 1 over eps int o w u lambda int bd o v u where u is a scalar density function and w and v are double well potentials we show that the behaviour of f e in the limit e 0 and depends on the limit of e log if this limit is finite and strictly positive then the singular limit of the energies f e leads to a coupled problem of bulk and surface phase transitions and under certain assumptions agrees with the relaxation of the capillary energy with line tension these results were announced in abs1 and abs2

on measuring sensitivity to parametric model misspecification summary in settings where parametric inference is inconsistent under model misspecification the discrepancy between correct and misspecified inferences is compared with the discrepancy between correct and misspecified models to make the comparison tractable large sample and small misspecification approximations are employed the ratio of the approximate discrepancy between inferences to the approximate discrepancy between models is regarded as a relative measure of sensitivity to model misspecification the maximum ratio over a family of correct distributions is determined as a measure of worst case sensitivity as well the distribution producing this maximum can be examined to see how a particular combination of a parametric family and estimand is susceptible to model misspecifications statisticians recognize that in reality no models are really correct in many parametric situations model misspecifications give rise to asymptotically inconsistent estimators of parameters the goal of this work is to quantify how bad an estimator can be relative to the degree of model misspecification the rationale for such an investigation is that misleading inferences arise when a bad answer is produced from an only mildly misspecified model we are less concerned about bad inferences arising from grossly misspecified models as such models are likely to be screened out by diagnostic checks we quantify the sensitivity of a parametric family and estimand to model misspecifications by comparing the kullback leibler kl divergence between correct and misspecified inferences with the kl divergence between correct and misspecified models more specifically the divergence between inferences can be based on either sampling distributions of maximum likelihood estimators or posterior distributions for tractability local approx imations to the kl divergence are employed in a somewhat similar spirit to cook 1986 the maximum ratio of approximate divergences over a class of correct models can be com puted as a worst case measure of sensitivity to misspecifications as well the correct model achieving the maximum can be examined to gain insight into the kind of departures from the model which are damaging this methodology is described fully in the next section then section 3 details an application to some common parametric models for survival data the paper concludes with a discussion in section 4 some technical comments are relegated to appendix a and appendix b

on inconsistent bayes estimates of location on montre que dans certaines situations relativement naturelles les estimateurs de position bayesiens sont inconsistants

on the consistency of bayes estimates on etudie les proprietes de frequence des regles de bayes avec une attention particuliere a la consistence

model for line tension in three phase equilibrium we propose a model free energy functional of two order parameters with which to calculate the interfacial and line tensions in three phase equilibrium the euler lagrange equations for the free energy minimum are solved exactly yielding the spatial variation of the order parameters analytically in terms of a parameter b2 in the model the three interfacial tensions in dimensionless form are 12 1 b2 12 1 b2 and 2 when b2 3 the three phases play symmetrical roles and the line tension again in the appropriate units is calculated to be 6 2 3 0 755 a wetting transition where the sum of two of the interfacial tensions becomes equal to the third occurs as b2 1 a quantity that approximates the line tension is found to vanish proportionally to the first power of the vanishing contact angle as the wetting transition is approached

on nearly optimal controls of hybrid lqg problems the paper is concerned with hybrid control of a class of linear quadratic gaussian lqg systems modulated by a finite state markov chain it develops approximation schemes for systems involving singularly perturbed markov chains with weak and strong interactions which are useful for natural time scale separation and for large scale markovian systems the presentation encompasses three cases the recurrent markov chains inclusion of transient states and inclusion of absorbing states near optimality of the approximation schemes is obtained and a numerical example is presented computational results indicate our approximation schemes perform well

robust jump linear quadratic control a mode stabilizing solution this correspondence is concerned with optimal feedback control of a class of linear stochastic systems for a quadratic performance index the solution to this problem is known but the resulting optimal regulator does not ensure stability in every mode of operation to improve robustness a new mode stabilizing solution is defined an example illustrates the advantages of the proposed controller

bayesian inference for semiparametric regression using a fourier representation summary this paper presents the bayesian analysis of a semiparametric regression model that consists of parametric and nonparametric components the nonparametric component is represented with a fourier series where the fourier coefficients are assumed a priori to have zero means and to decay to 0 in probability at either algebraic or geometric rates the rate of decay controls the smoothness of the response function the posterior analysis automatically selects the amount of smoothing that is coherent with the model and data posterior probabilities of the parametric and semiparametric models provide a method for testing the parametric model against a non specific alternative the bayes estimator s mean integrated squared error compares favourably with the theoretically optimal estimator for kernel regression

consistent and robust bayes procedures for location based on partial information we consider bayes procedures for a location parameter that are robust with respect to the shape of the distribution f of the data the case where f is fixed and the case where f has a dirichlet distribution are both treated the procedures are based on the posterior distributions of the location parameter given the partial information contained in a robust estimate of location we show consistency and asymptotic normality of the procedures and give instances where the bayes procedure based on the full sample diverges while the bayes procedures based on partial information converges and is asymptotically normal

maximum principle for optimal control problems of forward backward regime switching system and applications abstract in this paper we derive the stochastic maximum principle for optimal control problems of the forward backward markovian regime switching system the control system is described by forward backward sdes and modulated by continuous time finite state markov chains we first obtain the necessary and sufficient conditions for the optimal control thereafter we apply the maximum principle to recursive utility investment consumption problems and lq problems with markovian regime switching

nonlinear filtering and control of a switching diffusion with small observation noise this paper is concerned with nonlinear filtering and control of a switching diffusion coupled by an unknown markov chain two statistical estimation methods are used to track the unknown markov chain computable approximate filters are obtained based on these methods the filters are then used to construct controls for the partially observed system these controls are shown to be asymptotically optimal as the observation noise tends to zero finally an example is considered and numerical experiments are reported

nonparametric bayes estimators based on beta processes in models for life history data the article studies the problem of finding bayes estimators for cumulative hazard rates and related quantities w r t prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments a particular class of prior processes termed beta processes is introduced and is shown to constitute a conjugate class an important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated for example time inhomogeneous markov chains we find posterior distributions and derive bayes estimators in such models and present a semiparametric bayesian analysis of the cox regression model

optional polya tree and bayesian inference we introduce an extension of the polya tree approach for constructing distributions on the space of probability measures by using optional stopping and optional choice of splitting variables the construction gives rise to random measures that are absolutely continuous with piecewise smooth densities on partitions that can adapt to fit the data the resulting optional polya tree distribution has large support in total variation topology and yields posterior distributions that are also optional polya trees with computable parameter values

higher energy solutions in the theory of phase transitions a variational approach abstract we establish the existence of a higher energy solution to the vector ginzburg landau equation with a triple well potential on a bounded and smooth domain on the plane this solution is obtained by a linking argument in implementing this variational approach we make several considerations on the dynamics of the negative gradient flow in particular we use the conley index to contruct a suitable one dimensional invariant set this solution has morse index two in the nondegenerate case we discuss its structure in connection with the so called triple junction configurations

bayesian nonparametric estimation of the median part i computation of the estimates let x i i 1 ldots n be i i d sim f theta where f theta x f x theta for some f that has median equal to 0 f is assumed unknown or only partially known and the problem is to estimate theta priors are put on the pair f theta the priors on f concentrate all their mass on c d f s with median equal to 0 these priors include dirichlet type priors the marginal posterior distribution of theta given x 1 ldots x n is computed the mean of the posterior is taken as the estimate of theta

adaptive estimation of the threshold point in threshold regression this paper studies semiparametric efficient estimation of the threshold point in threshold regression the classical literature of semiparametric efficient estimation rests on the fact that the maximum likelihood estimator is efficient in any parametric submodel for a large class of loss functions however in threshold regression the maximum likelihood estimator is not efficient while the bayes estimators are efficient and different loss functions induce different efficient estimators for an additively separable loss function that separates the efficiency problem of the threshold point from that of other parameters we show that the semiparametric and parametric efficiency risk bounds coincide then we design a semiparametric empirical bayes estimator to achieve this bound in consequence the threshold point can be adaptively estimated even under conditional moment restrictions we also provide a valid confidence interval called the nonparametric posterior interval for the threshold point simulation studies show that the semiparametric empirical bayes approach is substantially better than existing methods to illustrate our procedure in practice we apply it to an economic growth model for detecting different growth patterns

misspecification in infinite dimensional bayesian statistics we consider the asymptotic behavior of posterior distributions if the model is misspecified given a prior distribution and a random sample from a distribution p 0 which may not be in the support of the prior we show that the posterior concentrates its mass near the points in the support of the prior that minimize the kullback leibler divergence with respect to p 0 an entropy condition and a prior mass condition determine the rate of convergence the method is applied to several examples with special interest for infinite dimensional models these include gaussian mixtures nonparametric regression and parametric models

bayesian nonparametric estimation for incomplete data via successive substitution sampling in the problem of estimating an unknown distribution function f in the presence of censoring one can use a nonparametric estimator such as the kaplan meier estimator or one can consider parametric modeling there are many situations where physical reasons indicate that a certain parametric model holds approximately in these cases a nonparametric estimator may be very inefficient relative to a parametric estimator on the other hand if the parametric model is only a crude approximation to the actual model then the parametric estimator may perform poorly relative to the nonparametric estimator and may even be inconsistent the bayesian paradigm provides a reasonable framework for this problem in a bayesian approach one would try to put a prior distribution on f that gives most of its mass to small neighborhoods of the entire parametric family we show that certain priors based on the dirichlet process prior can be used to accomplish this for these priors the posterior distribution of f given the censored data appears to be analytically intractable we provide a method for approximating this posterior distribution through the use of a successive substitution sampling algorithm we also show convergence of the algorithm

stability of hybrid dynamic systems containing singularly perturbed random processes to meet the challenge of handling dynamic systems in which continuous dynamics and discrete events coexist we develop stability analysis of such systems governed by ordinary differential equations and stochastic differential equations with regime switching modulated by a markov chain involving a small parameter the small parameter is used to reflect either the in herent two time scales in the original system or the different rates of regime switching among a large number of states of the discrete events the smaller the parameter is the more rapid switching the system will experience to reduce the complexity one attempts to effectively replace the actual systems by a limit system with simpler structure to ensure the validity of such a replacement in a longtime horizon it is crucial that the original system is stable the fast regime changes and the large state space of the markov chain make the stability analysis difficult under suitable conditions using the limit dynamic systems and the perturbed lyapunov func tion methods we show that if the limit systems are stable then so are the original systems this justifies the replacement of a complicated original system by its limit from a longtime behavior point of view index terms hybrid system lyapunov method markov chain singular perturbation stability switching diffusion

paper optimal control of markov chains admitting strong and weak interactions this paper is devoted to the study of an optimal control problem for a markov chain with generator b ea where e is a small parameter it is shown that an approximate solution can be calculated by a policy improvement algorithm involving computations relative to an aggregated problem the dimension of which is given by n the number of ergodic sets for the b matrix together with a family of decentralized problems the dimensions of which are given by the number of elements in each ergodic set for the b matrix

asymptotically optimal controls of hybrid linear quadratic regulators in discrete time this work develops asymptotically optimal controls for a class of discrete time hybrid systems involving singularly perturbed markov chains having weak and strong interactions the state space of the underlying markov chain is decomposed into a number of recurrent classes and a group of transient states using a hierarchical control approach by aggregating the states in each recurrent class into a single state a continuous time quadratic limit control problem in which the resulting limit markov chain has much smaller state space is derived using the optimal control for the limit problem a control for the original problem is constructed which is shown to be nearly optimal finally a numerical example is given to demonstrate the effectiveness of the approximation scheme

a representation of exchangeable hierarchies by sampling from random real trees a hierarchy on a set s also called a total partition of s is a collection mathcal h of subsets of s such that s in mathcal h each singleton subset of s belongs to mathcal h and if a b in mathcal h then a cap b equals either a or b or varnothing every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h associated as follows with a random real tree mathcal t equipped with root element 0 and a random probability distribution p on the borel subsets of mathcal t given mathcal t p let t 1 t 2 ldots be independent and identically distributed according to p and let mathcal h comprise all singleton subsets of mathbb n and every subset of the form j t j in f x as x ranges over mathcal t where f x is the fringe subtree of mathcal t rooted at x there is also the alternative characterization every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h derived as follows from a random hierarchy mathscr h on 0 1 and a family u j of i i d uniform 0 1 random variables independent of mathscr h let mathcal h comprise all sets of the form j u j in b as b ranges over the members of mathscr h

near optimal controls of random switching lq problems with indefinite control weight costs in this paper we consider hybrid controls for a class of linear quadratic problems with white noise perturbation and markov regime switching where the regime switching is modeled by a continuous time markov chain with a large state space and the control weights are indefinite the use of large state space enables us to take various factors of uncertain environment into consideration yet it creates computational overhead and adds difficulties aiming at reduction of complexity we demonstrate how to construct near optimal controls first in the model we introduce a small parameter to highlight the contrast of the weak and strong interactions and fast and slow motions this results in a two time scale formulation in view of the recent developments on lq problems with indefinite control weights and two time scale markov chains we then establish the convergence of the system of riccati equations associated with the hybrid lq problem based on the optimal feedback control of the limit system obtained using the system of riccati equations we construct controls for the original problem and show that such controls are near optimal a numerical demonstration of a simple system is presented here

numerical approximation of invariant measures for hybrid diffusion systems this work is concerned with numerical approximation of hybrid diffusions with regime switching modulated by continuous time finite state markov chains when using the euler maruyama approximation algorithms an important question is suppose the measure of the regime switching diffusions converges to its invariant measure will the sequence of measures of the approximation scheme converges to the same limit this paper provides answers to this question by appropriate interpolations and weak convergence methods it shows that a suitably interpolated sequence resulted from the algorithm converges to the switching diffusion the convergence to the invariant measure of the numerical algorithm is also studied

consistency of bayes estimators of a binary regression function when do nonparametric bayesian procedures overfit to shed light on this question we consider a binary regression problem in detail and establish frequentist consistency for a certain class of bayes procedures based on hierarchical priors called uniform mixture priors these are defined as follows let v be any probability distribution on the nonnegative integers to sample a function f from the prior v first sample m from v and then sample f uniformly from the set of step functions from 0 1 into 0 1 that have exactly m jumps i e sample all m jump locations and m 1 function values independently and uniformly the main result states that if a data stream is generated according to any fixed measurable binary regression function f o 1 2 then frequentist consistency obtains that is for any v with infinite support the posterior of v concentrates on any l1 neighborhood of f 0 solution of an associated large deviations problem is central to the consistency proof

finite horizon robust state estimation for uncertain finite alphabet hidden markov models with conditional relative entropy constraints we consider a robust state estimation problem for time varying uncertain discrete time homogeneous first order finite state finite alphabet hidden markov models hmms a class of time varying uncertain hmms is considered in which the uncertainty is sequentially described by a conditional relative entropy constraint on perturbed conditional probability measures given a realized observation sequence for this class of uncertain hmms the robust state estimation problem is formulated as a constrained optimization problem using a lagrange multiplier technique and a variational formula for conditional relative entropy the above problem is converted into an unconstrained optimization problem and a problem related to partial information risk sensitive filtering a measure transformation technique and an information state method are employed to solve this equivalent problem related to risk sensitive filtering a characterization of the solution to the robust state estimation problem is also presented

toward a reconciliation of the bayesian and frequentist approaches to point estimation abstract the bayesian and frequentist approaches to point estimation are reviewed the status of the debate regarding the use of one approach over the other is discussed and its inconclusive character is noted a criterion for comparing bayesian and frequentist estimators within a given experimental framework is proposed the competition between a bayesian and a frequentist is viewed as a contest with the following components a random observable a true prior distribution unknown to both statisticians an operational prior used by the bayesian a fixed frequentist rule used by the frequentist and a fixed loss criterion this competition is studied in the context of exponential families conjugate priors and squared error loss the class of operational priors that yield bayes estimators superior to the best frequentist estimator is characterized the implications of the existence of a threshold separating the space of operational priors into good and bad priors are explored and their relevance in ar

optimal adaptive lqg control for systems with finite state process parameters the situation where a totally observed process y t is generated by a stochastic differential equation whose parameters evolve on a finite set 1 n according to a stochastic differential equation is considered the optimal control law is sought with respect to quadratic loss functions on y t and the control u t the auxiliary p d e technique of hijab 6 is used together with a nonlinear filter to obtain the solution whose existence depends upon that of a smooth solution to the auxiliary p d e and strong solutions to the system s d e under the given control inputs

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

on inconsistent bayes estimates in the discrete case consider sampling from an unknown probability distribution on the integers with a tail free prior the posterior distribution is consistent with a mixture of a tail free prior and a point mass however the posterior may be inconsistent this is likewise true for a countable mixture of tail free priors similar results are given for dirichlet priors

jump linear quadratic gaussian control in continuous time the optimal quadratic control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes is addressed the systems are also subject to gaussian input and measurement noise the optimal solution for the jump linear quadratic gaussian jlqc problem is given this solution is based on a separation theorem the optimal state estimator is sample path dependent if the plant parameters are constant in each value of the underlying jumping process then the controller portion of the compensator converges to a time invariant control law however the filter portion of the optimal infinite time horizon jlqc compensator is not time invariant thus a suboptimal filter which does converge to a steady state solution under certain conditions is derived and a time invariant compensator is obtained

a reduction process for perturbed markov chains given a perturbed markov chain with generator a varepsilon a 0 varepsilon a 1 varepsilon 2 a 2 cdots it is shown that to each time scale t t varepsilon t varepsilon 2 cdots is associated a reduced markov chain this process of reduction has only finitely many steps it is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage it is shown that if lambda is an eigenvalue different from zero of the generator of the kth reduced chain with multiplicity p then a varepsilon admits p eigenvalues lambda varepsilon such that lambda varepsilon varepsilon k lambda o varepsilon k a basis of ker a 0 resp ker a 0 which is robust to the perturbation is constructed

the role of beliefs in inference for rational expectations models abstract this paper discusses inference for rational expectations models estimated via minimum distance methods by characterizing the probability beliefs regarding the data generating process dgp that are compatible with given moment conditions the null hypothesis is taken to be rational expectations and the alternative hypothesis to be distorted beliefs this distorted beliefs alternative is analyzed from the perspective of a hypothetical semiparametric bayesian who believes the model and uses it to learn about the dgp this interpretation provides a different perspective on estimates test statistics and confidence regions in large samples particularly regarding the economic significance of rejections in rational expectations models a suggestive application to the equity premium puzzle is examined

bayesian nonparametric estimation of the median part ii asymptotic properties of the estimates for data theta varepsilon i i 1 ldots n where varepsilon i are i i d sim f with the median of f equal to 0 but f otherwise unknown it is desired to estimate theta in doss 1985 priors are put on the pair f theta the marginal posterior distribution of theta is computed and the mean of the posterior is taken as the estimate of theta in the present paper a frequentist point of view is adopted the consistency properties of the bayes estimates computed in doss 1985 are investigated when the prior on f is of the dirichlet type any f whose median is 0 is in the support of these priors it is shown that if the varepsilon i are i i d from a discrete distribution then the bayes estimates are consistent however if the distribution of the varepsilon is is continuous the bayes estimates can be inconsistent

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

bsdes with regime switching weak convergence and applications abstract this paper is concerned with a system of backward stochastic differential equations bsdes with regime switching the bsdes are coupled by a finite state markov chain the underlying markov chain is assumed to have a two time scale or weak and strong interactions structure namely the states of the markov chain can be divided into a number of groups so that the chain jumps rapidly within a group and slowly between the groups it is shown in this paper that the original bsde system can be approximated by a limit system in which the states in each group are aggregated out and replaced by a single state in particular it is proved that the solution of the original bsde system converges weakly under the meyer zheng topology as the fast jump rate goes to infinity the limit process is a solution of aggregated bsdes which can be determined by the corresponding martingale problem the results are applied to a set of partial differential equations and used to validate their convergence to the corresponding limit system finally a numerical example is given to demonstrate the approximation results