existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters abstract the existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved together with a partial regularity result

a rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pdes we obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to lipschitz continuous viscosity solutions of uniformly elliptic partial differential equations 2007 wiley periodicals inc

efficient estimation of population level summaries in general semiparametric regression models this article considers a wide class of semiparametric regression models in which interest focuses on population level quantities that combine both the parametric and the nonparametric parts of the model special cases in this approach include generalized partially linear models generalized partially linear single index models structural measurement error models and many others for estimating the parametric part of the model efficiently profile likelihood kernel estimation methods are well established in the literature here our focus is on estimating general population level quantities that combine the parametric and nonparametric parts of the model e g population mean probabilities etc we place this problem in a general context provide a general kernel based methodology and derive the asymptotic distributions of estimates of these population level quantities showing that in many cases the estimates are semiparametric efficient for estimating the population mean with no missing data we s

on the permissible arrangements of ritz values for normal matrices in the complex plane abstract this article discusses ritz and harmonic ritz values computed from a krylov subspace generated by a normal matrix we give necessary and sufficient conditions for a given tuple of complex numbers to represent ritz values the existence of a positive solution to a certain linear system with a cauchy matrix using this characterization we prove several localization results for the ritz values of normal matrices and discuss consequences for the convergence of the restarted arnoldi algorithm similar results are shown for the harmonic case as well

asymptotics for kernel estimate of sliced inverse regression to explore nonlinear structures hidden in high dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression li and duan proposed the sliced inverse regression sir method which is simple to use in this paper the asymptotic properties of the kernel estimate of sliced inverse regression are investigated it turns out that regardless of the kernel function the asymptotic distribution remains the same for a wide range of smoothing parameters

efficiency loss caused by linearity condition in dimension reduction linearity sometimes jointly with constant variance is routinely assumed in the context of sufficient dimension reduction it is well understood that when these conditions do not hold blindly using them may lead to inconsistency in estimating the central subspace and the central mean subspace surprisingly we discover that even if these conditions do hold using them will bring efficiency loss this paradoxical phenomenon is illustrated through sliced inverse regression and principal hessian directions the efficiency loss also applies to other dimension reduction procedures we explain this empirical discovery by theoretical investigation

isoperimetry and stability properties of balls with respect to nonlocal energies we obtain a sharp quantitative isoperimetric inequality for nonlocal s perimeters uniform with respect to s bounded away from 0 this allows us to address local and global minimality properties of balls with respect to the volume constrained minimization of a free energy consisting of a nonlocal s perimeter plus a non local repulsive interaction term in the particular case s 1 the s perimeter coincides with the classical perimeter and our results improve the ones of knuepfer and muratov comm pure appl math 66 7 1129 1162 2013 comm pure appl math 2014 concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term more precisely their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

near sphere lattices with constant nonlocal mean curvature we are concerned with unbounded sets of mathbb r n whose boundary has constant nonlocal or fractional mean curvature which we call cnmc sets this is the equation associated to critical points of the fractional perimeter functional under a volume constraint we construct cnmc sets which are the countable union of a certain bounded domain and all its translations through a periodic integer lattice of dimension m le n our cnmc sets form a c 2 branch emanating from the unit ball alone and where the parameter in the branch is essentially the distance to the closest lattice point thus the new translated near balls or near spheres appear from infinity we find their exact asymptotic shape as the parameter tends to infinity

a paradox concerning nuisance parameters and projected estimating functions summary this paper is concerned with a paradox associated with parameter estimation in the presence of nuisance parameters in a statistical model with unknown nuisance parameters the efficiency of an estimator of a parameter usually increases when the nuisance para meters are known however the opposite phenomenon can sometimes occur in this paper we elucidate the occurrence of this paradox by examining estimating functions in particular we focus on the projected estimating function which is defined by the projection of the score function on to a given estimating function a sufficient condition for the paradox to occur is the orthogonality of the two components of the projected estimating functions corresponding to parameters of interest and nuisance parameters in addition a numerical assessment is conducted in the context of a simple model to investigate the improvement of the asymptotic efficiency of estimators

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

dimension reduction for non elliptically distributed predictors second order methods summary many classical dimension reduction methods especially those based on inverse conditional moments require the predictors to have elliptical distributions or at least to satisfy a linearity condition such conditions however are too strong for some applications li and dong 2009 introduced the notion of the central solution space and used it to modify first order methods such as sliced inverse regression so that they no longer rely on these conditions in this paper we generalize this idea to second order methods such as sliced average variance estimation and directional regression in doing so we demonstrate that the central solution space is a versatile framework we can use it to modify essentially all inverse conditional moment based methods to relax the distributional assumption on the predictors simulation studies and an application show a substantial improvement of the modified methods over their classical counterparts

weak and strong uniform consistency of kernel regression estimates we study the estimation of a regression function by the kernel method under mild conditions on the window the bandwidth and the underlying distribution of the bivariate observations x i y i we obtain the weak and strong uniform convergence rates on a bounded interval these results parallel those of silverman 1978 on density estimation and extend those of schuster and yakowitz 1979 and collomb 1979 on regression estimation

dimension reduction in regressions through cumulative slicing estimation in this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction in parallel to the classical slicing estimation we develop three methods that are termed respectively as cumulative mean estimation cumulative variance estimation and cumulative directional regression the strong consistency for p o n1 2 log n and the asymptotic normality for p o n1 2 are established where p is the dimension of the predictors and n is sample size such asymptotic results improve the rate p o n1 3 in many existing contexts of semiparametric modeling in addition we propose a modified bic type criterion to estimate the structural dimension of the central subspace its consistency is established when p o n1 2 extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration

sufficient dimension reduction through discretization expectation estimation in the context of sufficient dimension reduction the goal is to parsimoniously recover the central subspace of a regression model many inverse regression methods use slicing estimation to recover the central subspace the efficacy of slicing estimation depends heavily upon the number of slices however the selection of the number of slices is an open and long standing problem in this paper we propose a discretization expectation estimation method which avoids selecting the number of slices while preserving the integrity of the central subspace this generic method assures root n consistency and asymptotic normality of slicing estimators for many inverse regression methods and can be applied to regressions with multivariate responses a bic type criterion for the dimension of the central subspace is proposed comprehensive simulations and an illustrative application show that our method compares favourably with existing estimators

nonlocal minimal graphs in the plane are generically sticky we prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities more precisely we show that if a nonlocal minimal graph in a slab is continuous up to the boundary then arbitrarily small perturbations of the far away data produce boundary discontinuities hence either a nonlocal minimal graph is discontinuous at the boundary or a small perturbation of the prescribed conditions produces boundary discontinuities the proof relies on a sliding method combined with a fine boundary regularity analysis based on a discontinuity smoothness alternative namely we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is holder continuous up to the boundary in this spirit we prove that the boundary regularity of nonlocal minimal graphs in the plane jumps from discontinuous to c 1 gamma with no intermediate possibilities allowed in particular we deduce that the nonlocal curvature equation is always satisfied up to the boundary as an interesting byproduct of our analysis one obtains a detailed understanding of the switch between the regime of continuous and hence differentiable nonlocal minimal graphs to that of discontinuous and hence with differentiable inverse ones

regularity results for nonlocal equations by approximation we obtain c1 regularity estimates for nonlocal elliptic equations that are not necessarily translation invariant using compactness and perturbative methods and our previous regularity results for the translation invariant case

a semiparametric approach to dimension reduction we provide a novel and completely different approach to dimension reduction problems from the existing literature we cast the dimension reduction problem in a semiparametric estimation framework and derive estimating equations viewing this problem from the new angle allows us to derive a rich class of estimators and obtain the classical dimension reduction techniques as special cases in this class the semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact the common assumption of linearity and or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression the semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example this article has online supplementary material

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat