On some estimates based on sample behavior near high level excursions

Summary Let { _j } be a stationary sequence of weakly dependent random variables and letM _n ^(k) be thek-th largest value of _j , 1jn. The estimation of the parameters of the asymptotic distribution ofM _n ^(k) is considered using a procedure motivated by a limit theorem pertaining to the point process . A number of statistical issues concerning the procedure, including how to select the tuning parameters, are addressed. The second problem that we consider is the estimation of the filter of a moving average process with heavy tails. In particular, the investigation covers the moving average stable process. Motivated by ideas in Rootzén (1978), our estimator uses information contained in the sample behavior of the process near the largest excursion.Research supported by AFOSR Contract No. 91-0030, NAVY-ONR Grant No. N00014-92-J-1007, and NSF Grant No. 9107507     

The type of group measure space von Neumann algebras

LetG be a countable discrete group acting by measure-preserving automorphisms of a finite measure space (M, ) and let (G,M) be the corresponding group measure space von Neumann algebra, which will be a finite von Neumann algebra. Necessary and sufficient conditions are given for (G,M) to have a non-zero type I part, and the projection on the type I part is explicitly described.This research was supported in part by National Science Foundation Grant MCS 74-19876.     

Reliability bounds in DFRA class with known mean and variance

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Sub-Riemannian homogeneous spaces in dimensions 3 and 4

Let (M, , g) be a sub-Riemannian manifold (i.e. M is a smooth manifold, is a smooth distribution on M and g is a smooth metric defined on ) such that the dimension of M is either 3 or 4 and is a contact or odd-contact distribution, respectively. We construct an adapted connection on M and use it to study the equivalence problem. Furthermore, we classify the 3-dimensional sub-Riemannian manifolds which are sub-homogeneous and show the relation to Cartan's list of homogeneous CR manifolds. Finally, we classify the 4-dimensional sub-Riemannian manifolds which are sub-symmetric.     

A companion ideal of a multiplication module

Let M _R be a faithful multiplication module, where R is a commutative ring. As defined by Anderson, this ideal has proved to be useful in studying multiplication modules. First of all a cancellation law involving M and the ideals contained in is proved. Among various applications given, the following result is proved:: There exists a canonical isomorphism from onto such that for any ( Hom _R(M,M), x ( M, a ( (M), (xa) = x.(()(a). As an application of this later result it is proved that M is quasi-injective if and only if (M) is quasi-injective.     

Existence of polarized f-structures on collapsed manifolds with bounded curvature and diameter

We study the class of collapsed Riemanniann-manifolds with bounded sectional curvature and diameter. Our main result asserts that there is a constant,(n, d)>0, such that if a compactn-manifold has bounded curvature, , bounded diameter, diam (M ⁿ )d and sufficiently small volume, Vol(M ⁿ )(n, d), then it admits a mixed polarized F-structure. As a consequence, inf_g Vol(M ⁿ ,g)=0, where the infimum is taken over all metrics with . This assertion can be viewed as a weakened version of Gromov's critical volume conjecture.The first author is partially supported by NSF Grant DMS 9303999, and the second author is partially supported by NSF Grant DMS 9204095.     

Identifying nonlinear covariate effects in semimartingale regression models

Summary LetX _t be a semimartingale which is either continuous or of counting process type and which satisfies the stochastic differential equationdX _t=Y_t(t, Z_t) dt+dM_t, whereY andZ are predictable covariate processes,M is a martingale and is an unknown, nonrandom function. We study inference for by introducing an estimator for and deriving a functional central limit theorem for the estimator. The asymptotic distribution turns out to be given by a Gaussian random field that admits a representation as a stochastic integral with respect to a multiparameter Wiener process. This result is used to develop a test for independence ofX from the covariateZ, a test for time-homogeneity of , and a goodness-of-fit test for the proportional hazards model (t,z)=₁(t)a ₂(z) used in survival analysis.Research supported by the Army Research Office under Grant DAAL03-86-K-0094Research supported by the Air Force Office of Scientific Research under Contract F49620-85-C-0007     

Path processes and historical superprocesses

Summary A superprocessX over a Markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles.The historical process for is the process whose state at timet is the path of over time interval [0,t]. The superprocess over the historical superprocess over —reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process is that is a function of for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.Partially supported by National Science Foundation Grant DMS-8802667     

Rigidity of spheres in Riemannian manifolds and a non-embedding theorem

Letf:M be an isometric immersion between Riemannian manifolds. The purpose of this paper is to find the minimum possible conditions onM and (in the terms of curvatures and external diameter) in order to the image off be contained in a sphere. Our results generalize the other authors work in three major steps, domain, range and the codimension of immersions. As a byproduct, we obtain the non-embedding theorems Chern-Kuiper, Moore and Jacobowitz. The proofs are based on the maximum (comparison) principle.The author was supported by FAPEMIG Grant CEX-946/98.     

An affine characterization of the Veronese surface

IfM ² is a nondegenerate surface in a 4-dimensional Riemannian manifold , then there is a natural affine metricg defined onM ². It is shown that this affine metricg is conformal to the induced Riemannian metric onM ² if and only ifM ² is a minimal submanifold of in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ⁴ or hyperbolic spaceH ⁴ whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM ² is a nondegenerate surface in the standard 4-sphereS ⁴ whose affine metric is homothetic to the induced Riemannian metric, thenM ² is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961.     

Multiplicative posets

A functionf from the posetP to the posetQ is a strict morphism if for allx, y P withx we havef(x). If there is such a strict morphism fromP toQ we writeP Q, otherwise we writeP Q. We say a posetM is multiplicative if for any posetsP, Q withP M andQ M we haveP ×Q M. (Here (p ₁,q ₁)<(p ₂,q ₂) if and only ifp ₁<p ₂ andq ₁<q ₂.) This paper proves that well-founded trees with height are multiplicative posets.This research was supported in part by NSERC Grant #69-1325.     

Deformation of a submanifold in an Euclidean space with fixed Gauss image

In a previous paper [4] the present author studied a C mapping where M is an m-dimensional C manifold, I is some interval and for each tI the mapping is an immersion satisfying the following conditions, (i) The Gauss map associated with the immersion is regular. (ii) The Gauss image of the immersed submanifold is fixed against t for each point p of M. Such a mapping was called an admissible deformation. The purpose of the present paper is to give results obtained since then.     

Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator , it is possible to iterate and compute a sequence of estimates , using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh ^2km ,k=1, ...,t. Hereh ^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.     

Optimal global rates of convergence ofM-estimates for multivariate nonparametric regression

Consider the nonparametric regression modelY=go(T)+u, whereY is real-valued,u is a random error,T is a randomd-vector of explanatory variables ranging over a nondegenerated-dimensional compact setC, andgo(·) is the unknown smooth regression function, which ism (0) times continuously differentiable and itsmth partial derivatives satisfy the Hölder condition with exponent(0,1], wherei ₁, ...,i _d are nonnegative integers satisfying _k ^=1/d i _k =m. The piecewise polynomial estimator ofgo based onM-estimates is considered. It is proved that the rate of convergence of the underlying estimator is under certain regular conditions, which is the optimal global rate of convergence of least square estimates for nonparametric regression studied in [10–11].This work is partly supported by the National Natural Science Foundation of China.     

Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .     

Linear extension majority cycles on partial orders

Let P define a partial order on a set X of cardinalityn. A linear extensionL ofP is a linear order withP G L, and is the set of all linear extensions ofP. denotes that subset of withxLy forx, y X. A linear extension majority (LEM) relationM onX is defined byxMy if . Similarly,M is defined byxMy if . An LEM cycle exists if there arex, y, z X withxMyMzMx, and an LEM quasi-cycle exists ifxMyMzMx and the equality part of the definition ofM holds for exactly one pair in the triple. The study shows that no semiorders have LEM cycles or LEM quasi-cycles, and that every interval order has a maximal element under theM relation. LEM cycles and LEM quasi-cycles are also considered for partial orders with specific structures. Simulation is used to determine the relative likelihood with which LEM cycles and LEM quasi-cycles are observed when connected partial orders are generated at random by a specific procedure.Dr. Gehrlein's research was supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

Rigidity of minimal hypersurfaces of spheres with two principal curvatures

Let M be a compact oriented minimal hypersurface of the unit n-dimensional sphere Sⁿ. It is known that if the norm squared of the second fundamental form, , satisfies that for all , then M is isometric to a Clifford minimal hypersurface ([2], [5]). In this paper we will generalize this result for minimal hypersurfaces with two principal curvatures and dimension greater than 2. For these hypersurfaces we will show that if the average of the function is n - 1, then M must be a Clifford hypersurface. Received: 24 December 2002     

Modules Which Satisfy the Radical Formula

Let be a commutative ring. Then an -module M satisfies the radical formula when is a direct sum of a submodule M ₁ which satisfies the radical formula and a semi-artinian submodule M ₂.     

Totally Geodesic Subsets in the Manifold of Directions of Physical Space

Let M ₀ be the Minkowski 4-space, let ₂(M ₀) denote the second exterior power of M ₀ equipped with a structure of a pseudo-Euclidean space with signature (3,3), let K ₀(M ₀) ₂ M ₀ be the light cone, and let G _{1 2}(M ₀) be the set of the oriented 2-planes meeting the interior of K ₀(M ₀). Four types of totally geodesic two-manifolds in G ₁ are described such that manifolds of one type are pairwise congruent as subsets in ₂(M ₀), while manifolds of different types are not. Models of such manifolds in the disk D ³ are constructed. An explicit formula for the curvature tensor of G ₁ is given. Bibliography: 6 titles.