uniform rectifiability and harmonic measure ii poisson kernels in l p imply uniform rectifiability the first author was supported by nsf grants dms 0801079 and dms 1101244 the second author was supported by micinn grant mtm2010 16518 and by csic pie 200850i015 the third author was partially supported by grants dms 0901524 career dms 1056965 us nsf sloan research foundation and mtm2010 16232 mtm2009 14694 c02 01 spain

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

the lp neumann problem for the heat equation in non cylindrical domains we consider the lp neumann problem for the heat equation when 1 p 2 in a time varying domain using a perturbation technique and previous l2 neumann results the authors obtain a generalization of earlier work in caloric cylinders

boundary value problems in morrey spaces for elliptic systems on lipschitz domains let q be a bounded lipschitz domain in rn n 3 let be a second order elliptic system with constant coefficients satisfying the legendre hadamard condition we consider the dirichlet problem u 0 in q u f on 4 where e 0 depends on q and 0 a 2 for n 3 we obtain existence and uniqueness results with nontangential maximal function estimate u 2 a q c f 2 a 0q if c satisfies the strong elliptic condition and 0 a min n 1 2 e we show that the neumann type problem u 0 in q g g 2 a dq on 9q vu wfp idci has a unique solution here h2 x dq is an atomic space with the property h2 x dq l2 a 9q the invertibility of layer potentials on l2 x dq and h2 x dq is also obtained finally we study the dirichlet problem for the biharmonic equation we establish a similar estimate in l2 x for the biharmonic equation in which case the range 0 a 2 is sharp for n 4 or 5

a free boundary problem for the parabolic poisson kernel abstract we study parabolic chord arc domains introduced by hofmann lewis and nystrom 14 and prove a free boundary regularity result below the continuous threshold more precisely we show that a reifenberg flat parabolic chord arc domain whose poisson kernel has logarithm in vmo must in fact be a vanishing chord arc domain i e satisfies a vanishing carleson measure condition this generalizes to the parabolic setting a result of kenig and toro 26 and answers in the affirmative a question left open in the aforementioned paper of hofmann et al a key step in this proof is a classification of flat blowups for the parabolic problem

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

estimating a bounded normal mean the problem of estimating a normal mean has received much attention in recent years if one assumes however that the true mean lies in a bounded interval the problem changes drastically in this paper we show that if the interval is small approximately two standard deviations wide then the bayes rule against a two point prior is the unique minimax estimator under squared error loss for somewhat wider intervals we also derive sufficient conditions for minimaxity of the bayes rule against a three point prior

generalized linear quadratic control we consider the problem of stochastic finite and infinite horizon linear quadratic control under power constraints the calculations of the optimal control law can be done off line as in the classical linear quadratic gaussian control theory using dynamic programming which turns out to be a special case of the new theory developed in this technical note a numerical example is solved using the new methods

area integral estimates for caloric functions we study the relationship between the area integral and the para bolic maximal function of solutions to the heat equation in domains whose boundary satisfies a 5 1 mixed lipschitz condition our main result states that the area integral and the parabolic maximal function are equivalent in lp p 0 p oo the measure p must satisfy muckenhoupt s loo condition with respect to caloric measure we also give a fatou theorem which shows that the existence of parabolic limits is a e with respect to caloric measure equivalent to the finiteness of the area integral

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

weighted estimates in l 2 for laplace s equation on lipschitz domains let c r d d 3 be a bounded lipschitz domain for laplace s equation au 0 in q we study the dirichlet and neumann problems with boundary data in the weighted space l 2 leader where q q q 0 q 0 is a fixed point on and da denotes the surface measure on we prove that there exists e e 0 2 such that the dirichlet problem is uniquely solvable if 1 d d 3 e and the neumann problem is uniquely solvable if 3 d e a d 1 if q is a c 1 domain one may take e 2 the regularity for the dirichlet problem with data in the weighted sobolev space l 2 1 d is also considered finally we establish the weighted l 2 estimates with general ap weights for the dirichlet and regularity problems

efficient detection of random coefficients in autoregressive models the problem of detecting randomness in the coefficients of an ar p model that is the problem of testing ordinary ar p dependence against the alternative of a random coefficient autoregressive rcar p model is considered a nonstandard lan property is established for rcar p models in the vicinity of ar p ones two main problems arise in this context the first problem is related to the statistical model itself gaussian assumptions are highly unrealistic in a nonlinear context and innovation densities should be treated as nuisance parameters the resulting semiparametric model however appears to be severely nonadaptive in contrast with the linear arma case pseudo gaussian likelihood methods here are invalid under non gaussian densities even the innovation variance cannot be estimated without a strict loss of efficiency this problem is solved using a general result by hallin and werker which provides semiparametrically efficient central sequences without going through explicit tangent space calculations the second problem is related to the fact that the testing problem under study is intrinsically one sided while the case of multiparameter one sided alternatives is not covered by classical asymptotic theory under lan a concept of locally asymptotically most stringent somewhere efficient test is proposed in order to cope with this one sided nature of the problem

simple covariance approach to mathcal h _ infty analysis we revisit the classical mathcal h infty analysis problem of computing the ell 2 induced norm and the mathcal l 2 induced norm of a linear time invariant system for the discrete time case and the continuous time case respectively we follow an approach based on converting the problem of maximization over signals to that of maximization over a sort of deterministic covariance matrices the reformulation in terms of these covariance matrices greatly simplifies the dynamic analysis problem and converts the computation to a convex constrained matrix maximization problem furthermore the equivalence is for the actual mathcal h infty norm of the system rather than a bound and thus does not require the typical gamma iterations we argue that this approach is both attractive elementary and constructive in that the worst case disturbance is also easily obtained as a state feedback constructed from the solution of the matrix problem

counterexamples and uniqueness for lp ω oblique derivative problems abstract harmonic functions defined in lipschitz domains of the plane that have gradient nontangentially in l 2 and have nonnegative oblique derivative almost everywhere on the boundary with respect to a continuous transverse vector field are shown to be constant explicit examples that have almost everywhere vanishing oblique derivative are constructed when l 2 is replaced by l p p 2 explicit examples with vanishing oblique derivative are constructed when p 2 and the continuous vector field is replaced by large perturbations of the normal vector field optimal bounds on the perturbation depending on p 2 and the lipschitz constant are given which imply that only the constant solution has nonnegative oblique derivative almost everywhere examples are constructed in higher dimensions and the fredholm properties of certain nonvariational layer potentials discussed

on the asymptotic information bound on discute plusieurs resultats de borne inferieure pour la performance asymptotique d estimateurs de fonctionnelles regulieres dans des modeles independants et identiquement distribues l idee principale est de regarder un ensemble de lois limites locales d une suite d estimateurs plutot que d imposer des conditions de regularite ou de considerer des limites de risque maximum on accorde une attention particuliere a des situations ou le cone tangent n est pas un espace lineaire on calcule le risque minimax asymptotique local dans des modeles melanges

asymptotics for statistical treatment rules this paper develops asymptotic optimality theory for statistical treatment rules in smooth parametric and semiparametric models manski 2000 2002 2004 and dehejia 2005 have argued that the problem of choosing treatments to maximize social welfare is distinct from the point estimation and hypothesis testing problems usually considered in the treatment eects literature and advocate formal analysis of decision procedures that map empirical data into treatment choices we develop large sample approximations to statistical treatment assignment problems in both randomized experiments and observational data settings in which treatment eects are identified we derive a local asymptotic minmax regret bound on social welfare and a local asymptotic risk bound for a two point loss function we show that certain natural treatment assignment rules attain these bounds

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

a random linear functional approach to efficiency bounds abstract this paper suggests a method for obtaining efficiency bounds in models containing either only infinite dimensional parameters or both finite and infinite dimensional parameters semiparametric models the method is based on a theory of random linear functionals applied to the gradient of the log likelihood functional and is illustrated by computing the lower bound for cox s regression model

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

asymptotic properties of maximum likelihood estimates in the mixed poisson model this paper considers the asymptotic behavior of the maximum likelihood estimators mle s of the probabilities of a mixed poisson distribution with a nonparametric mixing distribution the vector of estimated probabilities is shown to converge in probability to the vector of mixed probabilities at rate n 1 2 varepsilon for any varepsilon 0 under a generalized chi 2 distance function it is then shown that any finite set of the mle s has the same joint limiting distribution as does the corresponding set of sample proportions when the support of the mixing distribution g 0 is an infinite set with a known upper bound and g 0 satisfies a certain condition at zero

efficient shrinkage in parametric models this paper introduces shrinkage for general parametric models we show how to shrink maximum likelihood estimators towards parameter subspaces de ned by general nonlinear restrictions we derive the asymptotic distribution and risk of the generalized shrinkage estimator using a local asymptotic framework we show that if the shrinkage dimension exceeds two the asymptotic risk of the shrinkage estimator is strictly less than that of the mle this reduction holds globally in the parameter space we show that the reduction in asymptotic risk is substantial even for moderately large values of the parameters the formula simplify in a very convenient way in the context of high dimensional models we derive a simple bound for the asymptotic risk we also provide a new large sample minimax e ciency bound we use the concept of local asymptotic minimax bounds a generalization of the conventional asymptotic minimax bounds the

minimax estimation of the mean of a normal distribution when the parameter space is restricted if x is a n theta 1 random variable let rho m be the minimax risk for estimation with quadratic loss subject to theta leq m then rho m 1 pi 2 m 2 o m 2 we exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions the approximate least favorable distributions correct to order m 2 have density m 1 cos 2 big frac pi 2m s big s leq m rather than the naively expected uniform density on lbrack m m rbrack we also show how our results extend to estimation of a vector mean and give some explicit solutions

overidentification in regular models in the unconditional moment restriction model of hansen specification tests and more efficient estimators are both available whenever the number of moment restrictions exceeds the number of parameters of interest we show that a similar relationship between potential refutability of a model and existence of more efficient estimators is present in much broader settings specifically a condition we name local overidentification is shown to be equivalent to both the existence of specification tests with nontrivial local power and the existence of more efficient estimators of some smooth parameters in general semi nonparametric models under our notion of local overidentification various locally nontrivial specification tests such as hausman tests incremental sargan tests or optimally weighted quasi likelihood ratio tests naturally extend to general semi nonparametric settings we further obtain simple characterizations of local overidentification for general models of nonparametric conditional moment restrictions with possibly different conditioning sets the results are applied to determining when semi nonparametric models with endogeneity are locally testable and when nonparametric plug in and semiparametric two step gmm estimators are semiparametrically efficient examples of empirically relevant semi nonparametric structural models are presented

the nonparanormal semiparametric estimation of high dimensional undirected graphs recent methods for estimating sparse undirected graphs for real valued data in high dimensional problems rely heavily on the assumption of normality we show how to use a semiparametric gaussian copula or nonparanormal for high dimensional inference just as additive models extend linear models by replacing linear functions with a set of one dimensional smooth functions the nonparanormal extends the normal by transforming the variables by smooth functions we derive a method for estimating the nonparanormal study the method s theoretical properties and show that it works well in many examples

a fast algorithm to computer the h norm of a transfer function matrix abstract a fast algorithm is presented to compute the h norm of a transfer function matrix based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related hamiltonian matrix the norm is computed with guaranteed accuracy a very simple method to compute a rather close lower bound on the h norm is given

boundary layer methods for lipschitz domains in riemannian manifolds abstract we extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the laplace equation and certain other constant coefficient elliptic partial differential equations on lipschitz domains in euclidean space we treat the laplace operator on lipschitz domains in a manifold with c 1 metric tensor and study the dirichlet neumann and oblique derivative boundary problems

a simplified approach to computing efficiency bounds in semiparametric models using some standard hilbert space theory a simplified approach to computing efficiency bounds in semiparametric models is presented we use some interesting examples to illustrate this approach and also obtain some results which seem to be new to the literature