Multivariate k-nearest neighbor density estimates

Under appropriate assumptions, expressions describing the asymptotic behavior of the bias and variance of k-nearest neighbor density estimates with weight function w are obtained. The behavior of these estimates is compared with that of kernel estimates. Particular attention is paid to the properties of the estimates in the tail.     

On spectral estimation for a homogeneous random process on the circle

A homogeneous random process on the circle {X(P): P∈S} is a process whose mean is zero and whose covariance function depends only on the angular distance θ between the points, i.e. E{X(P)}≡0 and E{X(P)X(Q)}=R(θ). We assume that the homogeneous process X(P) is observed at a finite number of points, equally spaced on the circle. Given independent realizations of the process, we first propose unbiased estimates for the parameters of the aliased spectrum and for the covariance function. We assume further that the process is Gaussian. The exact distribution of the spectral estimates and the asymptotic distribution of the estimates of the covariance function are derived. Finally, it is shown that the estimates proposed are in fact the maximum likelihood estimates and that they have minimum variance in the class of unbiased estimates.     

A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients

In recent years, there has been considerable interest in extending the well-known Calderon-Zygmund estimates for the Laplacian to more general equations, in particular equations with highest order coefficients lying in the Sarason space VMO. In addition, the analogous estimates with Morrey spaces replacing Lebesgue spaces have been considered. These Morrey space estimates have been proved by refining the proofs for the L^p estimates. We shall show that the Morrey space estimates follow from the L^p estimates via an elementary argument which is very similar to that used by Campanato.     

Minimum information estimation of allocation models of different sizes

For allocation models consisting of n = 10, 6 and 4 equations, constrained generalized least squares coefficient estimates are compared with those obtained from the minimum information (MI) criterion in the sense of statistical information theory. The MI estimates are more efficient for n = 10. The bootstrap procedure is illustrated for the assessment of the variability of these estimates.     

On the existence of consistent estimates and tests

The paper concerns estimates of probability measures (=p-measures) determined from a countable number of independent realizations. The main results are:

1. For suitable topologies, the existence of a consistent sequence of estimates implies the existence of a sequence of estimates which is strongly consistent a. e. with respect to an arbitrary finite prior measure. In the case of a measurable parameter the existence of a consistent sequence of estimates implies strong consistency of the Bayes estimates for the quadratic loss function.
2. If a family ofp-measures is separable with respect to the supremum metric, there exists a sequence of strict estimates which is uniformly consistent with respect to the supremum metric on any totally bounded subset ofp-measures.
3. IfB is totally σ-bounded with respect to the supremum metric there exists a density-consistent sequence of estimates. IfB is separable with respect to the supremum metric there exists a sequence of estimates which is density consistent a. e. with respect to an arbitrary finite prior measure.
4. A sequence of uniformly consistent tests exists for all compact hypotheses in the case of an abstract sample space and for all weakly closed hypotheses in the case of a separable metric sample space.

     

Efficiencies for window estimates of the parameters of the Cauchy distribution

Higgins and Tichenor [Appl. Math. and Comp. 3 (1977), 113-126] considered “window estimates” of location and reciprocal scale parameters for a general class of distributions and showed them to be asymptotically efficient for the Cauchy distribution. In this study, efficiencies of these estimates for the Cauchy distribution are investigated for small and moderate sample sizes by Monte Carlo methods. For n?40, window estimates of location are nearly optimal, and for n?20, they compare favorably with other easy-to-compute estimates. Window estimates of reciprocal scale are very good even for small samples and are nearly optimal for n?10. Thus, window estimates appear to have high efficiency for moderate as well as large sample sizes. Approximate normality is also investigated. The estimate of location converges rapidly to normality, whereas the estimate of reciprocal scale does not.     

Conjugate function and the modulus of continuity of k-th order

The estimates of the partial moduli of continuity of k-th order (k≧3) of the conjugate functions of several variables are obtained in the space ?(T ⁿ ). The exactness of these estimates is established by proper examples.     

Global estimates of fundamental solutions for higher-order Schr?dinger equations

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr?dinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related L ^p ?CL ^q estimates on the Schr?dinger solution. These estimates extend known results from the literature and are sharp. This result was lately already generalized to a degenerate case (cf. [4]).     

A ∞ weight estimates for vector-valued commutators of multilinear fractional integral

In this paper, the authors get the Coifman type weighted estimates and weak weighted LlogL estimates for vector-valued generalized commutators of multilinear fractional integral with w ∈ A∞. Furthermore, both the boundedness of vector-valued multilinear frac- tional integral and the weak weighted LlogL estimates for vector-valued multilinear fractional integral are also obtained.     

On the accuracy of normal approximation

The aim of the present paper is to obtain estimates of the speed of convergence in the central limit theorem in R^k for variation distance valid when (truncated) pseudo-moments are small enough. Together with the integral type estimates of Bhattacharya and Sweeting [5,6] the results of this paper lead to the integral type estimates in terms of pseudo-moments. Similar (but somewhat less general) results were anounced in [1].     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-λ inequality with two parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all Lp spaces for 1<p<∞. Pointwise estimates are then replaced by appropriate localized Lp-Lq estimates. We obtain weighted Lp estimates for a range of p that is different from (1,∞) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.     

Sharp probability estimates for random walks with barriers

We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,∞) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided the fourth moment is finite.     

Potential function estimates for quasi-Einstein metrics

In this paper, we derive the growth pinching estimates for potential functions of τ-quasi-Einstein metrics on complete noncompact connected manifolds, based on the estimates for the scalar curvature and the using of the weighted measure comparison theorem. Our results show that the estimates for potential functions rely on the sign of constants λ and μ.     

On the rate of convergence of the alternating projection method in finite dimensional spaces

Using the results of Smith, Solmon, and Wagner [K. Smith, D. Solomon, S. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977) 1227-1270] and Nelson and Neumann [S. Nelson, M. Neumann, Generalizations of the projection method with application to SOR theory for Hermitian positive semidefinite linear systems, Numer. Math. 51 (1987) 123-141] we derive new estimates for the speed of the alternating projection method and its relaxed version in Rm. These estimates can be computed in at most O(m³) arithmetic operations unlike the estimates in papers mentioned above that require spectral information. The new and old estimates are equivalent in many practical cases. In cases when the new estimates are weaker, the numerical testing indicates that they approximate the original bounds in papers mentioned above quite well.     

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H ^∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ?. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H ^k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.     

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic equations t_∂±x_∂(cx_∂) with a coefficient of bounded variations. These estimates are obtained by approximating c by piecewise constant coefficients, cε, and passing to the limit in the Carleman estimates associated to the operators defined with cε. Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.     

Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems

We prove existence, uniqueness, regularity results and estimates describing the behavior (both for large and small times) of a solution u of some nonlinear parabolic equations of Leray-Lions type including the p-Laplacian. In particular we show how the summability of the initial datum u₀ and the value of p influence the behavior of the solution u, producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates.     

Estimates of perturbed Oseen semigroups and their applications to the Navier-Stokes system in ℝ n

For perturbed Oseen semigroups in ? ⁿ , we establish their power L _p ? L _q estimates. These estimates are used to prove the existence of small global solutions to perturbed nonlinear Oseen systems and also of estimates of their L _p -norms as t → ∞.     

Parabolic equations in time dependent Reifenberg domains

In this paper we define time dependent parabolic Reifenberg domains and study Lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains. The basic assumption is that the principal coefficients are of parabolic BMO space with small parabolic BMO seminorms. It is shown that Lp estimates hold for time dependent parabolic δ-Reifenberg domains.     

Green function estimates for relativistic stable processes in half-space-like open sets

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m−(m^2/α−Δ)^α/2) in half-space-like C^1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M∈(0,∞). When m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)^α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for X^m, which is uniform for all m∈(0,∞), holds for a large class of non-smooth open sets.