Approximate Representation of Bergman Submodules

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L_a^2(B_d) satisfies P_M-sum from i to ( M_(zi)P_MM*_(zi))≤c/(N + 1)P_M for some number c > 0, then there is a sequence {f_j } of multipliers and a positive number c such that c'P_M ≤sum from j to ( M_(fj)M*_(fj))≤ P_M, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p > d.

The Number of Triangles in Random Intersection Graphs

Let Tn be the number of triangles in the random intersection graph G(n,m,p).When the mean of Tn is bounded,we obtain an upper bound on the total variation distance between Tn and a Poisson distribution.When the mean of Tn tends to infinity,the Stein–Tikhomirov method is used to bound the error for the normal approximation of Tn with respect to the Kolmogorov metric.

Coverage Accuracy of Confidence Intervals in Nonparametric Regression

Point-wise confidence intervals for a nonparametric regression function with random design points are considered. The confidence intervals are those based on the traditional normal approximation and the empirical likelihood. Their coverage accuracy is assessed by developing the Edgeworth expansions for the coverage probabilities. It is shown that the empirical likelihood confidence intervals are Bartlett correctable.

CONFIDENCE INTERVALS FOR NONPARAMETRIC REGRESSION FUNCTIONS WITH MISSING DATA： MULTIPLE DESIGN CASE

This paper considers two estimators of θ= g（x） in a nonparametric regression model Y = g（x） ＋ ε（x∈ （0, 1）p） with missing responses： Imputation and inverse probability weighted esti- mators. Asymptotic normality of the two estimators is established, which is used to construct normal approximation based confidence intervals on θ.

Bootstrap Approximation to the Distribution of M-estimates in a Linear Model

The authors establish the approximations to the distribution of M-estimates in a linear model by the bootstrap and the linear representation of bootstrap M-estimation, and prove that the approximation is valid in probability 1. A simulation is made to show the effects of bootstrap approximation, randomly weighted approximation and normal approximation.

NORMAL APPROXIMATION FOR FINITEPOPULATION U-STATISTICS

In this article, the general central limit theorem and Berry-Esseen bounds for finite-populationU-statistics with degree m are established under the very weak conditions, These resultssubstantially improve those of Zhao and Chen's (1987).

Dielectric Breakdown Model for an Electrically Semi-Permeable Penny-Shaped Crack in Three-Dimensional Piezoelectric Media

The dielectric breakdown（DB） model for a penny-shaped crack under a semipermeable boundary condition in a three-dimensional piezoelectric medium is studied.An approximate analytical solution is derived by using the boundary integral equation with extended displacement discontinuity,and the corresponding boundary element method with double iterative approaches is developed to analyze the semi-permeable crack.The effect of electric boundary conditions on crack faces is discussed on the basis of DB model.By comparing the DB model with the polarization saturation（PS） model for different piezoelectric materials,some interesting phenomena related to the electric yielding zone and local J-integral are observed.

CONVERGENCE RATE OFDISTRIBUTIONS OF TRIMMED SUMS

The authors first derive the normal expansion of the joint density function of two orderstatistics from the uniform distribution and then, using the approximation, establish a wayto estimate the normal convergence rate for trimmed sums. For applications, the convergencerates for the intermediately trimmed sums and heavily trimmed surns are found out.

Non-uniform Berry–Esseen Bounds forWeighted U-Statistics and Generalized L-Statistics

Weighted U-statistics and generalized L-statistics are commonly used in statistical inference and their asymptotic properties have been well developed.In this paper sharp non-uniform Berry–Esseen bounds for weighted U-statistics and generalized L-statistic are established.

A RANDOMLY WEIGHTED ESTIMATE OF THE POPULATION MEAN

Consider the model (1.6), where eij (j=1,…,Ni, i=1,2,…) are i.i.d.with mean 0 andwriance 1.Introduce a randomly weighted estimate βn defined by (1.8).Assuming e11 ～N(0, 1)and,the paper gives a necessary and sufficient condition for βn to be a consistent estimateof β0,and under some further restrictions a normal approximation fo βn is established whichcan be used in constructing a large sample confidence interval of β0. Finally, in the non-normalcase a theorem about the consistency of βn is proved.

Estimates for Fourier Coefficients of Cusp Forms in Weight Aspect

Let f be a holomorphic Hecke eigenform of weight k for the modular groupΓ = SL2(Z) and let λf(n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate 1λf(n21) x2 k2(log(x + k))6n≤x is established, which improves the previous result.