Nonparametric Estimation of Regression Functions in Point Process Models

We prove that the empirical L ²-risk minimizing estimator over some general type of sieve classes is universally, strongly consistent for the regression function in a class of point process models of Poissonian type (random sampling processes). The universal consistency result needs weak assumptions on the underlying distributions and regression functions. It applies in particular to neural net classes and to radial basis function nets. For the estimation of the intensity functions of a Poisson process a similar technique yields consistency of the sieved maximum likelihood estimator for some general sieve classes. This revised version was published online in August 2006 with corrections to the Cover Date.     

On polynomial selection for the general number field sieve

The general number field sieve (GNFS) is the asymptotically fastest algorithm for factoring large integers. Its runtime depends on a good choice of a polynomial pair. In this article we present an improvement of the polynomial selection method of Montgomery and Murphy which has been used in recent GNFS records.

     

Ⅰ型区间删失情形下部分线性模型Sieve极大似然估计渐近性质

本文在响应变量的观测值为Ⅰ型区间删失数据的情形下,讨论部分线性模型Ｓｉｅｖｅ极大似然估计的渐近性质．在一定条件下证明了该估计具有强相合性;参数分量的估计具有渐近正态性,并且是渐近有效的;非参数分量估计达到了最优弱收敛速度．     

一种Sieve极大似然估计的渐近性质

该文针对部分线性模型,在响应变量的观测值为Ⅰ型区间删失数据的情形下,讨论Ｓｉｅｖｅ极大似然估计的渐近性质．用三角级数来构造Ｓｉｅｖｅ空间,在一定条件下证明了该估计具有强相合性;得到了该估计的弱收敛速度,并且非参数部分的估计达到了最优收敛速度;还算出了参数部分的信息界．     

A tauberian theorem on dirichlet series

A very short proof is given for a theorem that has applications to sieve methods. The general applications appear in an earlier paper by the author and N. C. Ankeny, where the theorem was utilized but not proved.     

Nonasymptotic approach to Bayesian semiparametric inference

The classical semiparametric Bernstein–von Mises (BvM) results is reconsidered in a non-classical setup allowing finite samples and model misspecication. We obtain an upper bound on the error of Gaussian approximation of the posterior distribution for the target parameter which is explicit in the dimension of the target parameter and in the dimension of sieve approximation of the nuisance parameter. This helps to identify the so called critical dimension p_n of the sieve approximation of the full parameter for which the BvM result is applicable. If the bias induced by sieve approximation is small and dimension of sieve approximation is smaller then critical dimension than the BvM result is valid. In the important i.i.d. and regression cases, we show that the condition “p_n²q/n is small”, where q is the dimension of the target parameter and n is the sample size, leads to the BvM result under general assumptions on the model.     

The k-subset sum problem over finite fields

The subset sum problem is an important theoretical problem with many applications, such as in coding theory, cryptography, graph theory and other fields. One of the many aspects of this problem is to answer the solvability of the k-subset sum problem. It has been proven to be NP-hard in general. However, if the evaluation set has some special algebraic structure, it is possible to obtain some good conclusions. Zhu, Wan and Keti proposed partial results of this problem over two special kinds of evaluation sets. We generalize their conclusions in this paper, and propose asymptotical results of the solvability of the k-subset sum problem by using estimates of additive character sums over the evaluation set, together with the Brun sieve and the new sieve proposed by Li and Wan. We also apply the former two examples as application of our results.     

Amplification arguments for large sieve inequalities

We give a new proof of the arithmetic large sieve inequality based on an amplification argument, and use a similar method to prove a new sieve inequality for classical holomorphic cusp forms. A sample application of the latter is also given.     

A General Polynomial Sieve

An important component of the index calculus methods for finding discrete logarithms is the acquisition of smooth polynomial relations. Gordon and McCurley (1992) developed a sieve to aid in finding smooth Coppersmith polynomials for use in the index calculus method. We discuss their approach and some of the difficulties they found with their sieve. We present a new sieving method that can be applied to any affine subspace of polynomials over a finite field.     

An improvement for the large sieve for square moduli

We establish a result on the large sieve with square moduli. These bounds improve recent results by S. Baier [S. Baier, On the large sieve with sparse sets of moduli, J. Ramanujan Math. Soc. 21 (2006) 279-295] and L. Zhao [L. Zhao, Large sieve inequality for characters to square moduli, Acta Arith. 112 (3) (2004) 297-308].     

Ⅱ型区间删失数据下广义部分线性模型的筛极大似然估计(英文)

This article concerded with a semiparametric generalized partial linear model （GPLM） with the type Ⅱ censored data. A sieve maximum likelihood estimator （MLE） is proposed to estimate the parameter component, allowing exploration of the nonlinear relationship between a certain covariate and the response function. Asymptotic properties of the proposed sieve MLEs are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. Moreover, the estimators of the unknown parameters are asymptotically normal and efficient, and the estimator of the nonparametric function has an optimal convergence rate.     

On variants of the larger sieve

Gallagher's larger sieve is a powerful tool, when dealing with sequences of integers that avoid many residue classes. We present and discuss various variants of Gallagher's larger sieve. This revised version was published online in June 2006 with corrections to the Cover Date.     

A higher rank Selberg sieve and applications

We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve. This allows us to derive a simplified proof of the Zhang and Maynard-Tao result on bounded gaps between primes. We also apply the sieve to other subsequences of the primes and obtain bounded gaps in various settings.     

Sieve Estimates via Neural Network for Strong Mixing Processes

In this paper, we obtain the convergence rates of sieve estimates for α-mixing strictly stationary processes in the special case of neural networks. When the entropy of sieve satisfies certain conditions, we establish the bounds of sieve estimate. Using the bounds we give finally convergence rate of sieve estimate via neural networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sieve M-estimation for semiparametric varying-coefficient partially linear regression model

This article considers a semiparametric varying-coefficient partially linear regression model.The semiparametric varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable.A sieve M-estimation method is proposed and the asymptotic properties of the proposed estimators are discussed.Our main object is to estimate the nonparametric component and the unknown parameters simultaneously.It is easier to compute and the required computation burden is much less than the existing two-stage estimation method.Furthermore,the sieve M-estimation is robust in the presence of outliers if we choose appropriate ρ( ).Under some mild conditions,the estimators are shown to be strongly consistent;the convergence rate of the estimator for the unknown nonparametric component is obtained and the estimator for the unknown parameter is shown to be asymptotically normally distributed.Numerical experiments are carried out to investigate the performance of the proposed method.     

Large Sieve Inequality with Quadratic Amplitudes

In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish non-trivial bounds.     

Large Sieve Inequality with Quadratic Amplitudes

In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish non-trivial bounds.     

Constructing quasisymmetric functions via graph-directed iterated function systems

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.     

Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the gaussian integer method

In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. We also illustrate our results by successfully computing discrete logarithms with GNFS in a large prime field.

     

An arithmetical examples of a non-Borel Luzin analytic set

N. N. Luzin has constructed a sieve defining a screened non-Borel analytic set and has indicated the arithmetical properties of points of this set. In this article it is shown that a screened analytic set contains continuum number of points which do not have these arithmetical properties. Necessary and sufficient arithmetical conditions on points are found, under which they lie in a Luzin set screened by a sieve.Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 525–534, October, 1977.The author thanks P. L. Ul'yanov for assistance with the article.