Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

分位点函数的光滑非参数估计的BAHADUR表示

文中对分位函数给出了具有更广泛应用的光滑分位估计,证明了该光滑分位估计的逐点和一致的Bahadur强表示定理;并由此结果推导了估计的重对数律,强逼近等深刻结果。     

On defining OLAP formulations

^** Email: vutsinas{at}upatras.gr Recently there has been increasing interest in On Line AnalyticalProcessing (OLAP) to satisfy the organizational needs of high-levelinformation delivery and advanced data analysis. The actualapplication of OLAP tools involves the use of various functions,such as the common drilling down and slicing and dicing. Usuallyeach particular OLAP function is comprehensive and intuitive.However, sophisticated use of OLAP tools requires complicatedcombinations of different OLAP functions that are not straight-forwardfor end users or designers. In this paper we attempt to enumerateand formally define OLAP functions by defining a new OLAP modelthat provides a broader view of OLAP. We demonstrate the expressiveadequacy of the new OLAP model with application examples.     

A Survey on Topological Properties of Tiles Related to Number Systems

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results on their fundamental group. If a self-affine tile tessellates the space the structure of the set of its ‘neighbors’ is discussed.     

On an additive representation function

Let A be an infinite subset of natural numbers, and X a positive real number. Let r(n) denotes the number of solution of the equation n=a₁+a₂ where a₁?a₂ and a₁, a₂∈A. Also let |A(X)| denotes the number of natural numbers which are less than or equal to X and belong to A. For those A which satisfy the condition that for all sufficiently large natural numbers n we have r(n)≠1, we improve the lower bound of |A(X)| given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible.     

Schopenhauer's Prolegomenon to Fuzziness

Prolegomenon means something said in advance of something else. In this study, we posit that part of the work by Arthur Schopenhauer (1788–1860) can be thought of as a prolegomenon to the existing concept of fuzziness. His epistemic framework offers a comprehensive and surprisingly modern framework to study individual decision making and suggests a bridgeway from the Kantian program into the concept of fuzziness, which may have had its second prolegomenon in the work by Frege, Russell, Wittgenstein, Peirce and Black. In this context, Zadeh's seminal contribution can be regarded as the logical consequence of the Kant-Schopenhauer representation framework.     

Infinite order differential operators in spaces of entire functions

Differential operators ?(Δ_θ,ω), where ? is an exponential type entire function of a single complex variable and Δ_θ,ω=(θ+ωz)D+zD², D=∂/∂z, , θ?0, , acting in the spaces of exponential type entire function are studied. It is shown that, for ω?0, such operators preserve the set of Laguerre entire functions provided the function ? also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane . The operator exp(aΔ_θ,ω), a?0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all . An integral representation of exp(aΔ_θ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ_θ,ω.     

Generalized Padé-Type Approximation and Integral Representations

In several complex variables, the multivariate Padé-type approximation theory is based on the polynomial interpolation of the multidimensional Cauchy kernel and leads to complicated computations. In this paper, we replace the multidimensional Cauchy kernel by the Bergman kernel function K (z,x) into an open bounded subset of C ⁿ and, by using interpolating generalized polynomials for K (z,x), we define generalized Padé-type approximants to any f in the space OL ²() of all analytic functions on which are of class L ². The characteristic property of such an approximant is that its Fourier series representation with respect to an orthonormal basis for OL ²() matches the Fourier series expansion of f as far as possible. After studying the error formula and the convergence problem, we show that the generalized Padé-type approximants have integral representations which give rise to the consideration of an integral operator – the so-called generalized Padé-type operator – which maps every f OL ²() to a generalized Padé-type approximant to f. By the continuity of this operator, we obtain some convergence results about series of analytic functions of class L ². Our study concludes with the extension of these ideas into every functional Hilbert space H and also with the definition and properties of the generalized Padé-type approximants to a linear operator of H into itself. As an application we prove a Painlevé-type theorem in C ⁿ and we give two examples making use of generalized Padé-type approximants.     

Random Menshov spectra

We show that the spectra of frequencies obtained by random perturbations of the integers allows one to represent any measurable function on by an almost everywhere converging sum of harmonics: