asymptotically efficient estimation of models defined by convex moment inequalities this paper examines the efficient estimation of partially identified models defined by moment inequalities that are convex in the parameter of interest in such a setting the identified set is itself convex and hence fully characterized by its support function we provide conditions under which despite being an infinite dimensional parameter the support function admits n consistent regular estimators a semiparametric efficiency bound is then derived for its estimation and it is shown that any regular estimator attaining it must also minimize a wide class of asymptotic loss functions in addition we show that the plug in estimator is efficient and devise a consistent bootstrap procedure for estimating its limiting distribution the setting we examine is related to an incomplete linear model studied in beresteanu and molinari 2008 and bontemps magnac and maurin 2012 which further enables us to establish the semiparametric efficiency of their proposed estimators for that problem

efficient estimation for the proportional hazards model with interval censoring the maximum likelihood estimator mle for the proportional hazards model with case 1 interval censored data is studied it is shown that the mle for the regression parameter is asymptotically normal with n convergence rate and achieves the information bound even though the mle for the baseline cumulative hazard function only converges at n 1 3 rate estimation of the asymptotic variance matrix for the mle of the regression parameter is also considered to prove our main results we also establish a general theorem showing that the mle of the finite dimensional parameter in a class of semiparametric models is asymptotically efficient even though the mle of the infinite dimensional parameter converges at a rate slower than n the results are iliustrated by applying them to a data set from a tumorigenicity study

a construction of a nonparametric quantum information manifold abstract we present a construction of a banach manifold structure on the set of faithful normal states of a von neumann algebra where the underlying banach space is a quantum analogue of an orlicz space on the manifold we introduce the exponential and mixture connections as dual pair of affine connections

identification and estimation of nonlinear dynamic panel data models with unobserved covariates abstract this paper considers nonparametric identification of nonlinear dynamic models for panel data with unobserved covariates including such unobserved covariates may control for both the individual specific unobserved heterogeneity and the endogeneity of the explanatory variables without specifying the distribution of the initial condition with the unobserved variables we show that the models are nonparametrically identified from two periods of the dependent variable y i t and three periods of the covariate x i t the main identifying assumptions include high level injectivity restrictions and require that the evolution of the observed covariates depends on the unobserved covariates but not on the lagged dependent variable we also propose a sieve maximum likelihood estimator mle and focus on two classes of nonlinear dynamic panel data models i e dynamic discrete choice models and dynamic censored models we present the asymptotic properties of the sieve mle and investigate the finite sample properties of these sieve based estimators through a monte carlo study an intertemporal female labor force participation model is estimated as an empirical illustration using a sample from the panel study of income dynamics psid

efficiency bounds for estimating linear functionals of nonparametric regression models with endogenous regressors abstract let y x e where is unknown and e e x 0 with positive probability but there exist instrumental variables w such that e e w 0 w p 1 it is well known that such nonparametric regression models are generally ill posed in the sense that the map from the data to is not continuous in this paper we derive the efficiency bounds for estimating certain linear functionals of without assuming itself to be identified

generalized correlation function definition properties and application to blind equalization with an abundance of tools based on kernel methods and information theoretic learning a void still exists in incorporating both the time structure and the statistical distribution of the time series in the same functional measure in this paper a new generalized correlation measure is developed that includes the information of both the distribution and that of the time structure of a stochastic process it is shown how this measure can be interpreted from a kernel method as well as from an information theoretic learning points of view demonstrating some relevant properties to underscore the effectiveness of the new measure a simple blind equalization problem is considered using a coded signal

information geometry on hierarchy of probability distributions an exponential family or mixture family of probability distributions has a natural hierarchical structure this paper gives an orthogonal decomposition of such a system based on information geometry a typical example is the decomposition of stochastic dependency among a number of random variables in general they have a complex structure of dependencies pairwise dependency is easily represented by correlation but it is more difficult to measure effects of pure triplewise or higher order interactions dependencies among these variables stochastic dependency is decomposed quantitatively into an orthogonal sum of pairwise triplewise and further higher order dependencies this gives a new invariant decomposition of joint entropy this problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections the orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families as an example we decompose the dependency in a higher order markov chain into a sum of those in various lower order markov chains

probability product kernels the advantages of discriminative learning algorithms and kernel machines are combined with generative modeling using a novel kernel between distributions in the probability product kernel data points in the input space are mapped to distributions over the sample space and a general inner product is then evaluated as the integral of the product of pairs of distributions the kernel is straightforward to evaluate for all exponential family models such as multinomials and gaussians and yields interesting nonlinear kernels furthermore the kernel is computable in closed form for latent distributions such as mixture models hidden markov models and linear dynamical systems for intractable models such as switching linear dynamical systems structured mean field approximations can be brought to bear on the kernel evaluation for general distributions even if an analytic expression for the kernel is not feasible we show a straightforward sampling method to evaluate it thus the kernel permits discriminative learning methods including support vector machines to exploit the properties metrics and invariances of the generative models we infer from each datum experiments are shown using multinomial models for text hidden markov models for biological data sets and linear dynamical systems for time series data

adaptive learning in complex reproducing kernel hilbert spaces employing wirtinger s subgradients this paper presents a wide framework for non linear online supervised learning tasks in the context of complex valued signal processing the complex input data are mapped into a complex reproducing kernel hilbert space rkhs where the learning phase is taking place both pure complex kernels and real kernels via the complexification trick can be employed moreover any convex continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response the only requirement is the subgradient of the adopted loss function to be available in an analytic form in order to derive analytically the subgradients the principles of the recently developed wirtinger s calculus in complex rkhs are exploited furthermore both linear and widely linear in rkhs estimation filters are considered to cope with the problem of increasing memory requirements which is present in almost all online schemes in rkhs the sparsification scheme based on projection onto closed balls has been adopted we demonstrate the effectiveness of the proposed framework in a non linear channel identification task a non linear channel equalization problem and a quadrature phase shift keying equalization scheme using both circular and non circular synthetic signal sources

an error entropy minimization algorithm for supervised training of nonlinear adaptive systems the paper investigates error entropy minimization in adaptive systems training we prove the equivalence between minimization of error s renyi 1970 entropy of order spl alpha and minimization of a csiszar 1981 distance measure between the densities of desired and system outputs a nonparametric estimator for renyi s entropy is presented and it is shown that the global minimum of this estimator is the same as the actual entropy the performance of the error entropy minimization criterion is compared with mean square error minimization in the short term prediction of a chaotic time series and in nonlinear system identification

geometry of deformed exponential families invariant dually flat and conformal geometries an information geometrical foundation is established for the deformed exponential families of probability distributions two different types of geometrical structures an invariant geometry and a flat geometry are given to a manifold of a deformed exponential family the two different geometries provide respective quantities such as deformed free energies entropies and divergences the class belonging to both the invariant and flat geometries at the same time consists of exponential and mixture families theq families are characterized from the viewpoint of the invariant and flat geometries the q exponential family is a unique class that has the invariant and flat geometries in the extended class of positive measures furthermore it is the only class of which the riemannian metric is conformally connected with the invariant fisher metric

efficient estimation of the partly linear additive cox model the partly linear additive cox model is an extension of the linear cox model and allows flexible modeling of covariate effects semiparametrically we study asymptotic properties of the maximum partial likelihood estimator of this model with right censored data using polynomial splines we show that with a range of choices of the smoothing parameter the number of spline basis functions required for estimation of the nonparametric components the estimator of the finite dimensional regression parameter is root n consistent asymptotically normal and achieves the semiparametric information bound rates of convergence for the estimators of the nonparametric components are obtained they are comparable to the rates in nonparametric regression implementation of the estimation approach can be done easily and is illustrated by using a simulated example

stochastic blind equalization based on pdf fitting using parzen estimator this work presents a new blind equalization approach that aims to force the probability density function pdf at the equalizer output to match the known constellation pdf quadratic distance between pdf s is used as the cost function to be minimized the proposed method relies on the parzen window method to estimate the data pdf and is implemented by a stochastic gradient descent algorithm the kernel size of the parzen estimator allows a dual mode switch or a soft switch between blind and decision directed equalization the proposed method converges faster than the constant modulus algorithm cma working at the symbol rate with a similar computational burden and reduces the residual error of the cma in multilevel modulations at the same time a comparison with the most common blind techniques is presented

bayesian inverse problems with gaussian priors the posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter and the smoothness and scale of the prior correct combinations of these characteristics lead to the minimax rate the frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter with smoother priors leading to zero coverage and rougher priors to conservative coverage in the latter case credible sets are of the correct order of magnitude the results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

sieve m inference on irregular parameters abstract this paper presents sieve inferences on possibly irregular i e slower than root n estimable functionals of semi nonparametric models with i i d data we provide a simple consistent variance estimator of the plug in sieve m estimator of a possibly irregular functional and the asymptotic standard normality of the sieve t statistic we show that for hypothesis testing of irregular functionals the sieve likelihood ratio statistic is asymptotically chi square distributed these results are useful in inference on structural parameters that may have singular semiparametric efficiency bounds a simulation study and an empirical application of heckman and singer 1984 duration model are presented

differentiability of statistical functionals and consistency of the jackknife in statistical applications the unknown parameter of interest can frequently be defined as a functional 0 t f where f is an unknown population statistical inferences about 0 are usually made based on the statistic t fn where fn is the empirical distribution assessing t fn as an estimator of 0 or making large sample inferences usually requires a consistent estimator of the asymptotic variance of t fn asymptotic behaviour of the jackknife variance estimator is closely related to the smoothness of the functional t this paper studies the smoothness of t through the differentiability of t and establishes some general results for the consistency of the jackknife variance estimators the results are applied to some examples in which the statistics t fn are l m estimators and some test statistics

wigner yanase information on quantum state space the geometric approach in the search of appropriate riemannian metrics on quantum state space the concept of statistical monotonicity or contraction under coarse graining has been proposed by chentsov the metrics with this property have been classified by petz all the elements of this family of geometries can be seen as quantum analogs of fisher information although there exists a number of general theorems shedding light on this subject many natural questions also stemming from applications are still open in this paper we discuss a particular member of the family the wigner yanase information using a well known approach that mimics the classical pull back approach to fisher information we are able to give explicit formulas for the geodesic distance the geodesic path the sectional and scalar curvatures associated to wigner yanase information moreover we show that this is the only monotone metric for which such an approach is possible

on entropy minimization and convergence we examine the minimization of information entropy for measures on the phase space of bounded domains subject to constraints that are averages of grand canonical distributions we describe the set of all such constraints and show that it equals the set of averages of all probability measures absolutely continuous with respect to the standard measure on the phase space with the exception of the measure concentrated on the empty configuration we also investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy we then show that for fixed constraints the parameters of the corresponding grand canonical distribution converge as volume increases to the corresponding parameters derivatives when they exist of the thermodynamic limit entropy the results hold when the energy is the sum of any stable tempered interaction potential that satisfies the gibbs variational principle e g lennard jones and the kinetic energy the same tools and the strict convexity of the thermodynamic limit pressure for continuous systems valid whenever the gibbs variational principle holds give solid foundation to the folklore local homeomorphism between thermodynamic and macroscopic quantities

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

likelihood based inference for current status data on a grid a boundary phenomenon and an adaptive inference procedure in this paper we study the nonparametric maximum likelihood estimator npmle for an event time distribution function at a point in the current status model with observation times supported on a grid of potentially unknown sparsity and with multiple subjects sharing the same observation time this is of interest since observation time ties occur frequently with current status data the grid resolution is specified ascn withc 0 being a scaling constant and 0 regulating the sparsity of the grid relative to the number of subjects n the asymptotic behavior falls into three cases depending on regular normal type asymptotics obtain for 1 3 and 1 3 serves as a boundary at which the transition happens the limit distribution at the boundary is different from either of the previous cases and converges weakly to those obtained with 2 0 1 3 and 2 1 3 1 asc goes to1 and 0 respectively this weak convergence allows us to develop an adaptive procedure to construct confidence intervals for the value of the event time distribution at a point of interest without needing to know or estimate which is of enormous advantage from the perspective of inference a simulation study of the adaptive procedure is presented

on profile likelihood abstract we show that semiparametric profile likelihoods where the nuisance parameter has been profiled out behave like ordinary likelihoods in that they have a quadratic expansion in this expansion the score function and the fisher information are replaced by the efficient score function and efficient fisher information the expansion may be used among others to prove the asymptotic normality of the maximum likelihood estimator to derive the asymptotic chi squared distribution of the log likelihood ratio statistic and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance

optimal convergence rates bahadur representation and asymptotic normality of partitioning estimators abstract this paper studies the asymptotic properties of partitioning estimators of the conditional expectation function and its derivatives mean square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions the uniform rate explicitly accounts for the effect of moment assumptions which is useful in semiparametric inference a general asymptotic integrated mean square error approximation is obtained and used to derive an optimal plug in tuning parameter selector a uniform bahadur representation is developed for linear functionals of the estimator using this representation asymptotic normality is established along with consistency of a standard error estimator the finite sample performance of the partitioning estimator is examined and compared to other nonparametric techniques in an extensive simulation study

efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals this paper considers semiparametric efficient estimation ofconditional moment models with possibly nonsmooth residuals in unknown parametric components and unknown functions h of endogenous variables we show that 1 the penalized sieve minimum distance psmd estima tor can simultaneously achieve root n asymptotic normality ofand nonparametric optimal convergence rate of allowing for noncompact function parameter spaces 2 a simple weighted bootstrap procedure consistently estimates the limiting distribution of the psmd 3 the semi parametric efficiency bound formula of ai and chen 2003 remains valid for conditional models with nonsmooth residuals and the optimally weighted psmd estimator achieves the bound 4 the centered profiled optimally weighted psmd criterion is asymptotically chi square distributed we illustrate our theories using a partially linear quantile instrumental variables iv regression a monte carlo study and an empirical estimation of the shape invariant quantile iv engel curves

a multigrid homotopy method for nonlinear inverse problems abstract in the present contribution we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems in which the forward problems are modeled by some forms of partial differential equations the method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest and then carries out the inversion on the coarsest grid by the homotopy method the strategy may give a rapidly and globally convergent method as a practical application this method is used to solve the nonlinear inverse problem of a nonlinear convection diffusion equation which is the saturation equation within the two phase porous media flow we demonstrate the effectiveness and merits of the multigrid homotopy method on two actual model problems

continuum versus discrete networks graph laplacians and reproducing kernel hilbert spaces abstract motivated by applications to machine learning we construct a reversible and irreducible markov chain whose state space is a certain collection of measurable sets of a chosen l c h space x we study the resulting network connected undirected graph including transience royden and riesz decompositions and kernel factorization we describe a construction for hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of l 2 functions by functions of finite energy the latter has applications to machine learning for markov random fields for example

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

likelihood ratio of unidentifiable models and multilayer neural networks this paper discusses the behavior of the maximum likelihood estimator mle in the case that the true parameter cannot be identified uniquely among many statistical models with unidentifiability neural network models are the main concern of this paper it is known in some models with unidentifiability that the asymptotics of the likelihood ratio of the mle has an unusually larger order using the framework of locally conic models put forth by dacunha castelle and gassiat as a generalization of hartigan s idea a useful sufficient condition of such larger orders is derived this result is applied to neural network models and a larger order is proved if the true function is given by a smaller model also under the condition that the model has at least two redundant hidden units a log n lower bound for the likelihood ratio is derived

semiparametric efficiency bound for models of sequential moment restrictions containing unknown functions abstract this paper computes the semiparametric efficiency bound for finite dimensional parameters identified by models of sequential moment restrictions containing unknown functions our results extend those of chamberlain 1992b and ai and chen 2003 for semiparametric conditional moment restrictions with identical information sets to the case of nested information sets and those of chamberlain 1992a and brown and newey 1998 for models of sequential moment restrictions without unknown functions to cases with unknown functions of possibly endogenous variables our results are applicable to semiparametric panel data models and two stage plug in problems as an important example we compute the efficiency bound for a weighted average derivative of a nonparametric instrumental variables regression npiv and find that simple plug in npiv estimators are not efficient we present an optimally weighted orthogonalized sieve minimum distance estimator that achieves the semiparametric efficiency bound

classical and quantum probability we follow the development of probability theory from the beginning of the last century emphasizing that quantum theory is really a generalization of this theory the great achievements of probability theory such as the theory of processes generalized random fields estimation theory and information geometry are reviewed their quantum versions are then described

efficient estimation of banach parameters in semiparametric models abtract consider a semiparametric model with a euclidean parameter and an infinite dimensional parameter to be called banach parameter assume there exists an efficient estimator of the euclidean parameter when the value of the euclidean parameter is known there exists an estimator of the banach parameter which depends on this value and is efficient within this restricted model substituting the efficient estimator of the euclidean parameter for the value of this parameter in the estimator of the banach parameter one obtains an efficient estimator of the banach parameter for the full semiparametric model with the euclidean parameter unknown this heredity property of efficiency completes estimation in semiparametric models in which the euclidean parameter has been estimated efficiently typically estimation of both the euclidean and the banach parameter is necessary in order to describe the random phenomenon under study to sufficient extent since efficient estimators are asymptotically linear the above substitution method is a particular case of substituting asymptotically linear estimators of a euclidean parameter into estimators that are asymptotically linear themselves and that depend on this euclidean parameter this more general substitution case is studied for its own sake as well and a heredity property for asymptotic linearity is proved

asymptotically optimal estimation of smooth functionals for interval censoring case 2 for a version of the interval censoring model case 2 in which the observation intervals are allowed to be arbitrarily small we consider estimation of functionals that are differentiable along hellinger differentiable paths the asymptotic information lower bound for such functionals can be represented as the squared l2 norm of the canonical gradient in the observation space this canonical gradient has an implicit expression as a solution of an integral equation that does not belong to one of the standard types we study an extended version of the integral equation that can also be used for discrete distribution functions like the nonparametric maximum likelihood estimator npmle and derive the asymptotic normality and efficiency of the npmle from properties of the solutions of the integral equations

intrinsic stochastic differential equations as jets we explain how ito stochastic differential equations sdes on manifolds may be defined using 2 jets of smooth functions we show how this relationship can be interpreted in terms of a convergent numerical scheme we also show how jets can be used to derive graphical representations of ito sdes and we show how jets can be used to derive the differential operators associated with sdes in a coordinate free manner we relate jets to vector flows giving a geometric interpretation of the ito stratonovich transformation we show how percentiles can be used to give an alternative coordinate free interpretation of the coefficients of one dimensional sdes we relate this to the jet approach this allows us to interpret the coefficients of sdes in terms of fan diagrams in particular the median of an sde solution is associated with the drift of the sde in stratonovich form for small times

semiparametric likelihood ratio inference likelihood ratio tests and related confidence intervals for a real parameter in the presence of an infinite dimensional nuisance parameter are considered in all cases the estimator of the real parameter has an asymptotic normal distribution however the estimator of the nuisance parameter may not be asymptotically gaussian or may converge to the true parameter value at a slower rate than the square root of the sample size nevertheless the likelihood ratio statistic is shown to possess an asymptotic chi squared distribution the examples considered are tests concerning survival probabilities based on doubly censored data a test for presence of heterogeneity in the gamma frailty model a test for significance of the regression coefficient in cox s regression model for current status data and a test for a ratio of hazards rates in an exponential mixture model in both of the last examples the rate of convergence of the estimator of the nuisance parameter is less than the square root of the sample size

an infinite dimensional geometric structure on the space of all the probability measures equivalent to a given one let m be the set of all probability densities equivalent to a given reference probability measure this set is thought of as the maximal regular i e with strictly positive densities dominated statistical model for each f m we define 1 a banach space l f with unit ball v f and 2 a mapping s f from a subset u f of m onto v f in such a way that the system s f u f f m is an affine atlas on m moreover each parametric exponential model dominated by is a finite dimensional affine submanifold and each parametric statistical model dominated by with a suitable regularity is a submanifold the global geometric framework given by the manifold structure adds some insight to the so called geometric theory of statistical models in particular the present paper gives some of the developments connected with the fisher information metrics rao and the hilbert bundle introduced by amari

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

localized mirror functor for lagrangian immersions and homological mirrorsymmetry for mathbb p 1_ a b c this paper gives a new way of constructing landau ginzburg mirrors using deformation theory of lagrangian immersions motivated by the works of seidel strominger yau zaslow and fukaya oh ohta ono moreover we construct a canonical functor from the fukaya category to the mirror category of matrix factorizations this functor derives homological mirror symmetry under some explicit assumptions as an application the construction is applied to spheres with three orbifold points to produce their quantum corrected mirrors and derive homological mirror symmetry furthermore we discover an enumerative meaning of the inverse mirror map for elliptic curve quotients

a homotopy method for well log constraint waveform inversion the primary difficulty posed by the method of waveform inversion for the acoustic wave equation is the presence of local minima of the objective function waveform inversion fails to converge to the global minimum unless the wavefield contains very low frequencies or the starting model is very close to the true model constraints can improve the convergence of the method however if the starting model is far from the correct model normal iterative methods will still get trapped in local minima we designed a homotopy method to improve the robustness of waveform inversion it makes natural use of constraints such as well logs in addition to further condition the inverse problem we incorporate standard tikhonov regularization we demonstrate through a synthetic example that our method is more likely to find a global minimum than normal iterative methods

using pilot data to size a two arm randomized trial to find a nearly optimal personalized treatment strategy a personalized treatment strategy formalizes evidence based treatment selection by mapping patient information to a recommended treatment personalized treatment strategies can produce better patient outcomes while reducing cost and treatment burden thus among clinical and intervention scientists there is a growing interest in conducting randomized clinical trials when one of the primary aims is estimation of a personalized treatment strategy however at present there are no appropriate sample size formulae to assist in the design of such a trial furthermore because the sampling distribution of the estimated outcome under an estimated optimal treatment strategy can be highly sensitive to small perturbations in the underlying generative model sample size calculations based on standard uncorrected asymptotic approximations or computer simulations may not be reliable we offer a simple and robust method for powering a single stage two armed randomized clinical trial when the primary aim is estimating the optimal single stage personalized treatment strategy the proposed method is based on inverting a plugin projection confidence interval and is thereby regular and robust to small perturbations of the underlying generative model the proposed method requires elicitation of two clinically meaningful parameters from clinical scientists and uses data from a small pilot study to estimate nuisance parameters which are not easily elicited the method performs well in simulated experiments and is illustrated using data from a pilot study of time to conception and fertility awareness

homotopy polynomial equations gross boundary data and small helmholtz systems inverse problems of the boundary measurement type appear in several geophysical contexts including dc resistivity electromagnetic induction and groundwater flow the objective is to determine a spatially varying coefficient in a partial differential equation from incomplete knowledge of the dependent variable and its normal gradient at the boundary equivalent 2d discrete inverse problems based on the helmholtz or modified helmholtz equation reduce to systems of polynomial equations indicating that there are only a finite number of exact solutions excluding certain pathological cases a homotopy procedure decides whether real positive solutions exist and if so generates the entire list the computational complexity of the algorithm scales asmm 2 wheremis the number of model parameters to be found measurement errors are accommodated by oversampling the boundary data at additional frequencies for test helmholtz and modified helmholtz inverse problems based on i perfect and ii noisy data i generate the full list of exact solutions the homotopy approach applies to large scale multidimensional geophysical inverse problems but at present is practical only for small systems up tom 9 recent advances in homotopy theory should however reduce the complexity making larger problems tractable in the future

a reproducing kernel hilbert space framework for information theoretic learning this paper provides a functional analysis perspective of information theoretic learning itl by defining bottom up a reproducing kernel hilbert space rkhs uniquely determined by the symmetric nonnegative definite kernel function known as the cross information potential cip the cip as an integral of the product of two probability density functions characterizes similarity between two stochastic functions we prove the existence of a one to one congruence mapping between the itl rkhs and the hilbert space spanned by square integrable probability density functions therefore all the statistical descriptors in the original information theoretic learning formulation can be rewritten as algebraic computations on deterministic functional vectors in the itl rkhs instead of limiting the functional view to the estimators as is commonly done in kernel methods a connection between the itl rkhs and kernel approaches interested in quantifying the statistics of the projected data is also established

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

measures of entropy from data using infinitely divisible kernels information theory provides principled ways to analyze different inference and learning problems such as hypothesis testing clustering dimensionality reduction classification and so forth however the use of information theoretic quantities as test statistics that is as quantities obtained from empirical data poses a challenging estimation problem that often leads to strong simplifications such as gaussian models or the use of plug in density estimators that are restricted to certain representation of the data in this paper a framework to nonparametrically obtain measures of entropy directly from data using operators in reproducing kernel hilbert spaces defined by infinitely divisible kernels is presented the entropy functionals which bear resemblance with quantum entropies are defined on positive definite matrices and satisfy similar axioms to those of renyi s definition of entropy convergence of the proposed estimators follows from concentration results on the difference between the ordered spectrum of the gram matrices and the integral operators associated to the population quantities in this way capitalizing on both the axiomatic definition of entropy and on the representation power of positive definite kernels the proposed measure of entropy avoids the estimation of the probability distribution underlying the data moreover estimators of kernel based conditional entropy and mutual information are also defined numerical experiments on independence tests compare favorably with state of the art

information geometry and sufficient statistics information geometry provides a geometric approach to families of statistical models the key geometric structures are the fisher quadratic form and the amari chentsov tensor in statistics the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information this leads to the question how the geometric structures behave under such sufficient statistics while this is well studied in the finite sample size case in the infinite case we encounter technical problems concerning the appropriate topologies here we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations within this framework we can then handle the topological issues and show that the fisher metric and the amari chentsov tensor on statistical models in the class of symmetric 2 tensor fields and 3 tensor fields can be uniquely up to a constant characterized by their invariance under sufficient statistics thereby achieving a full generalization of the original result of chentsov to infinite sample sizes more generally we decompose markov morphisms between statistical models in terms of statistics in particular a monotonicity result for the fisher information naturally follows

homological mirror symmetry for curves of higher genus abstract this paper is devoted to homological mirror symmetry conjecture for curves of higher genus it was proposed by katzarkov as a generalization of original kontsevich s conjecture a version of this conjecture in the case of the genus two curve was proved by seidel 25 based on the paper of seidel we prove the conjecture in the same version for curves of genus g 3 namely we relate the fukaya category of a genus g curve to the category of singularities of zero fiber in the mirror dual landau ginzburg model we also prove a kind of reconstruction theorem for hypersurface singularities namely formal type of hypersurface singularity i e a formal power series up to a formal change of variables can be reconstructed with some technical assumptions from its d z 2 g category of landau ginzburg branes the precise statement is theorem 1 2

a differential geometric approach to nonlinear filtering the projection filter this paper presents a new and systematic method of approximating exact nonlinear filters with finite dimensional filters using the differential geometric approach to statistics the projection filter is defined rigorously in the case of exponential families a convenient exponential family is proposed which allows one to simplify the projection filter equation and to define an a posteriori measure of the local error of the projection filter approximation finally simulation results are discussed for the cubic sensor problem

enumerative meaning of mirror maps for toric calabi yau manifolds abstract we prove that the inverse of a mirror map for a toric calabi yau manifold of the form k y where y is a compact toric fano manifold can be expressed in terms of generating functions of genus 0 open gromov witten invariants defined by fukaya oh ohta ono 2010 15 such a relation between mirror maps and disk counting invariants was first conjectured by gross and siebert 2011 24 conjecture 0 2 and remark 5 1 as part of their program and was later formulated in terms of fukaya oh ohta ono s invariants in the toric calabi yau case in chan et al 2012 8 conjecture 1 1

parametric rate inference for one sided differentiable parameters suppose one has a collection of parameters indexed by a possibly infinite dimensional set given data generated from some distribution the objective is to estimate the maximal parameter in this

an infinite dimensional statistical manifold modelled on hilbert space abstract we construct an infinite dimensional hilbert manifold of probability measures on an abstract measurable space the manifold m retains the first and second order features of finite dimensional information geometry the divergences admit first derivatives and mixed second derivatives enabling the definition of the fisher metric as a pseudo riemannian metric this is enough for many applications for example it justifies certain projections of markov processes onto finite dimensional submanifolds in recursive estimation problems m was constructed with the fenchel legendre transform between kullback leibler divergences and its role in bayesian estimation in mind this transform retains on m the symmetry of the finite dimensional case many of the manifolds of finite dimensional information geometry are shown to be c embedded submanifolds of m in establishing this we provide a framework in which many of the formal results of the finite dimensional subject can be proved with full rigour

an improved quantum projection filter abstract this work extends the previous quantum projection filtering scheme in gao et al 2019 by adding an optimality analysis result a reformulation of the quantum projection filter is derived by minimizing the truncated stratonovich stochastic taylor expansion of the difference between the true quantum trajectory and its approximation on a lower dimensional submanifold through quantum information geometric techniques simulation results for a qubit example demonstrate better approximation performance for the new quantum projection filter

a homotopy method for nonlinear inverse problems we develop a homotopy method for nonlinear inverse problems where the forward problems are governed by some forms of differential equations a tikhonov style regularization approach yields an optimization problem ordinary iterative methods may fail to solve this problem due to their locally convergent properties then the fixed point homotopy method is introduced to solving the normal equation of the optimization problem and a new and globally convergent algorithm is constructed which is highly effective in the aspects of speed of computation ability of noise suppression and wide region of convergence as a practical application the method is used to solve the inverse problem of 2 d acoustic wave equation we demonstrate the merits and effectiveness of our algorithm on two realistic model problems

minimax estimation of a functional on a structured high dimensional model we introduce a new method of estimation of parameters in semi parametric and nonparametric models the method is based on estimating equations that are u statistics in the observations the u statistics are based on higher order influence functions that extend ordinary linear influence functions of the parameter of interest and represent higher derivatives of this parameter for parameters for which the representation cannot be perfect the method leads to a bias variance trade off and results in estimators that converge at a slower than n rate in a number of examples the resulting rate can be shown to be optimal we are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity where the parameter of interest cannot be estimated at n rate but we also consider efficient n estimation using novel nonlinear estimators the general approach is applied in detail to the example of estimating a mean response when the response is not always observed

a q exponential statistical banach manifold abstract let be a given probability measure and m the set of equivalent strictly positive probability densities in this paper we construct a banach manifold on m modeled on the space l p where p is a reference density for the non parametric q exponential statistical models tsallis s deformed exponential where 0 q 1 is any real number this family is characterized by the fact that when q 1 then the non parametric exponential models are obtained and the manifold constructed by pistone and sempi is recovered up to continuous embeddings on the modeling space the coordinate mappings of the manifold are given in terms of csiszar s divergences the tangent vectors are identified with the one dimensional q exponential models and q deformations of the score function

lagrangian floer potential of orbifold spheres abstract for each sphere with three orbifold points we construct an algorithm to compute the open gromov witten potential which serves as the quantum corrected landau ginzburg mirror and is an infinite series in general this gives the first class of general type geometries whose full potentials can be computed as a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients furthermore we prove that the open gromov witten potential is convergent even in the general type cases and has an isolated singularity at the origin which is an important ingredient of proving homological mirror symmetry

gromov witten theory of orbicurves the space of tri polynomials and symplectic field theory of seifert fibrations we compute with symplectic field theory sft techniques the gromov witten theory of mathbb p 1 alpha 1 ldots alpha a i e the complex projective line with a orbifold points a natural subclass of these orbifolds the ones with polynomial quantum cohomology gives rise to a family of polynomial frobenius manifolds and integrable systems of hamiltonian pdes which extend the dispersionless bigraded toda hierarchy carlet the extended bigraded toda hierarchy arxiv preprint arxiv math ph 0604024 we then define a frobenius structure on the spaces of polynomials in three complex variables of the form f x y z xyz p1 x p2 y p3 z which contains as special cases the ones constructed on the space of laurent polynomials dubrovin geometry of 2d topologica field theories integrable systems and quantum groups springer lecture notes in mathematics 1620 120 348 1996 milanov and tseng the space of laurent polynomials mathbb p 1 orbifolds and integrable hierarchies preprint arxiv math 0607012v3 math ag we prove a mirror theorem stating that these frobenius structures are isomorphic to the ones found before for polynomial mathbb p 1 orbifolds finally we link rational sft of seifert fibrations over mathbb p 1 a b c with orbifold gromov witten invariants of the base extending a known result bourgeois a morse bott approach to contact homology ph d dissertation stanford university 2002 valid in the smooth case

estimation of a convex function characterizations and asymptotic theory we study nonparametric estimation of convexregression and density functions by methods of least squares in the regression and density cases and maximum likelihood in the density estimation case we provide characterizations of these estimators prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated the asymptotic distribution theory relies on the existence of an invelope function for integrated two sided brownian motion t 4 which is established in a companion paper by groeneboom jongbloed and wellner

nonexistence of finite dimensional filters for conditional statistics of the cubic sensor problem consider the cubic sensor dx dw dy x3dt dv where w v are two independent brownian motions given a function x of the state x let hat phi t left x right denote the conditional expection given the observations ys 0 s t this paper consists of a rather detailed discussion and outline of proof of the theorem that for nonconstant there can not exist a recursive finite dimensional filter for hat phi driven by the observations

riemann manifold langevin and hamiltonian monte carlo methods the paper proposes metropolis adjusted langevin and hamiltonian monte carlo sampling methods defined on the riemann manifold to resolve the shortcomings of existing monte carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations the methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for metropolis hastings or indeed hamiltonian monte carlo and metropolis adjusted langevin algorithms this allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the markov chain the methodology proposed exploits the riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold providing highly efficient convergence and exploration of the target density the performance of these riemann manifold monte carlo methods is rigorously assessed by performing inference on logistic regression models log gaussian cox point processes stochastic volatility models and bayesian estimation of dynamic systems described by non linear differential equations substantial improvements in the time normalized effective sample size are reported when compared with alternative sampling approaches matlab code that is available from http www ucl ac uk statistics research rmhmc allows replication of all the results reported

polynomial programming using multi homogeneous polynomial continuation abstract a polynomial programming problem is a nonlinear programming problem where the objective function inequality constraints and equality constraints all consist of polynomial functions the necessary optimality conditions for such a problem can be formulated as a polynomial system of equations among whose zeros the global optimum must lie this note applies the theory of multi homogeneous also called m homogeneous polynomials and recent homotopy algorithms to the polynomial system formulation of the necessary optimality conditions significantly reducing the work of a naive homotopy approach the m homogeneous homotopy approach providing the global optimum is practical for small problems for example the geometric modeling problem of finding the distance between two polynomial surfaces is a polynomial programming problem the theory is illustrated in the last section by this geometric modeling problem and a prototype structural design problem

information theoretic clustering clustering is an important topic in pattern recognition since only the structure of the data dictates the grouping unsupervised learning information theory is an obvious criteria to establish the clustering rule the paper describes a novel valley seeking clustering algorithm using an information theoretic measure to estimate the cost of partitioning the data set the information theoretic criteria developed here evolved from a renyi entropy estimator a renyi 1960 that was proposed recently and has been successfully applied to other machine learning applications j c principe et al 2000 an improved version of the k change algorithm is used in optimization because of the stepwise nature of the cost function and existence of local minima even when applied to nonlinearly separable data the new algorithm performs well and was able to find nonlinear boundaries between clusters the algorithm is also applied to the segmentation of magnetic resonance imaging data mri with very promising results

estimation of treatment effects in randomized trials with non compliance and a dichotomous outcome summary we propose a class of estimators of the treatment effect on a dichotomous outcome among the treated subjects within covariate and treatment arm strata in randomized trials with non compliance recent papers by vansteelandt and goetghebeur and robins and rotnitzky have presented consistent and asymptotically linear estimators of a causal odds ratio which rely beyond correct specification of a model for the causal odds ratio on a correctly specified model for a potentially high dimensional nuisance parameter in this paper we propose consistent asymptotically linear and locally efficient estimators of a causal relative risk and a new parameter called a switch causal relative risk which relies only on the correct specification of a model for the parameter of interest our estimators are always consistent and asymptotically linear at the null hypothesis of no treatment effect thereby providing valid testing procedures we examine the finite sample properties of these instrumental variable based estimators and the associated testing procedures in simulations and a data analysis of decaffeinated coffee consumption and miscarriage

a statistical manifold modeled on orlicz spaces using kaniadakis κ exponential models abstract given a probability measure on the space of strictly positive densities m we construct a topological manifold on which the elements are connected by exponential models in the form q exp u k p u p where exp x 1 2 x 2 x 1 x y x 1 2 y 2 y 1 2 x 2 p q m and their local representations are elements of an orlicz space i e the manifold is modeled on orlicz spaces parameter k is the g kaniadakis parameter for deformed exponentials which is strongly relevant to relativity and statistical complex models in statistical mechanics functional k p is the deformed counterpart of the cumulant mapping and satisfies that if 0 we obtain the usual cumulant functional of the exponential manifold moreover in this limit case the exponential manifold constructed by pistone and sempi is recovered in the context of deformed exponentials we prove that the function cosh 1 where cosh x is the deformed hyperbolic cosine is a young function and generates the orlicz space on which the exponential manifold is modeled namely l p this construction differs from the one made by pistone on the paper exponential models from the geometrical viewpoint since this last one is based on divergence functionals and modeled on lebesgue spaces l 1 p the use of deformed models is interesting since they generalize the exponential models and extend them to non additive systems