Consistency and normality of Huber-Dutter estimators for partial linear model

For partial linear model Y = Xτβ0 g0(T) with unknown β0 ∈ Rd and an unknown smooth function g0, this paper considers the Huber-Dutter estimators of β0, scale σ for the errors and the function g0 approximated by the smoothing B-spline functions, respectively. Under some regularity conditions, the Huber-Dutter estimators of β0 and σ are shown to be asymptotically normal with the rate of convergence n-1/2 and the B-spline Huber-Dutter estimator of g0 achieves the optimal rate of convergence in nonparametric regression. A simulation study and two examples demonstrate that the Huber-Dutter estimator of β0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator.     

The restricted EM algorithm under linear inequalities in a linear model with missing data

This paper discusses the maximum likelihood estimate of βunder linear inequalities A0β≥a in a linear model with missing data, proposes the restricted EM algorithm and proves the convergence.     

Weak consistency of least-squares estimators in linear models

Let Y_n, n≥1, be a sequence of integrable random variables with EY_n = x_n1β₁ + x_n2β₂ + … + x_npβ_p, where the x_ij's are known and β^T = (β₁, β₂,…, β_p) unknown. Let b_n be the least-squares estimator of β based on Y₁, Y₂,…, Y_n. Weak consistency of b_n, n≥1, has been considered in the literature under the assumption that each Y_n is square integrable. In this paper, we study weak consistency of b_n, n≥1, and associated rates of convergence under the minimal assumption that each Y_n is integrable.     

随机截断下部分线性模型中参数估计的渐近性质

考虑部分线性回归模型Yi＝xiβ g(ti) σiej,i＝1，2，…，n其中σi^2＝／f(ui)．当Yi因受某种随机干扰而被右截断时，就截断分布巳知的情形，利用所获得的截断观察数据构造了β，g，f的估计量β^～n，g^～n，f^～n，并在一定条件下，证明了β^～n的渐近正态性，同时得到了g^～n，f^～n的最优收敛速度。     

Convergence rates of MLE in a partly linear model

This paper considers the estimation for a partly linear model with case 1 interval censored data. We assume that the error distribution belongs to a known family of scale distributions with an unknown scale parameter. The sieve maximum likelihood estimator (MLE) for the model’s parameter is shown to be strongly consistent, and the convergence rate of the estimator is obtained and discussed.     

具有外生变量部分线性自回归模型的样条估计

考虑自回归模型Y_t=θ~TX_t g(Zt) ε_t,t=1,…,n,其中X_t=(Y_(t-1),…,Y_(t-d))~T,Z_t为实值外生随机变量,θ=(θ_1,…,θ_d)~T为待估参数向量,g为未知非参数光滑函数．基于多项式样条方法,在一定的条件下,给出了θ的估计的渐近正态性,得到了g的估计的收敛速度．模拟例子验证了所得的理论结果．     

Estimation of partial linear error-in-response models with validation data

In this paper, an estimation theory in partial linear model is developed when there is measurement error in the response and when validation data are available. A semiparametric method with the primary data is used to define two estimators for both the regression parameter and the nonparametric part using the least squares criterion with the help of validation data. The proposed estimators of the parameter are proved to be strongly consistent and asymptotically normaal, and the estimators of the nonparametric part are also proved to be strongly consistent and weakly consistent with an optimal convergent rate. Then, the two estimators of the parameter are compared based on their empirical performances. Supported by NNSF of China (No. 10231030, No. 10241001) and a grant to the author for his excellent Ph.D. dissertation work in China.     

Dimension reduction in partly linear error-in-response models with validation data

Consider partial linear models of the form Y=X^τβ+g(T)+e with Y measured with error and both p-variate explanatory X and T measured exactly. Let be the surrogate variable for Y with measurement error. Let primary data set be that containing independent observations on and the validation data set be that containing independent observations on , where the exact observations on Y may be obtained by some expensive or difficult procedures for only a small subset of subjects enrolled in the study. In this paper, without specifying any structure equations and distribution assumption of Y given , a semiparametric dimension reduction technique is employed to obtain estimators of β and g(·) based the least squared method and kernel method with the primary data and validation data. The proposed estimators of β are proved to be asymptotically normal, and the estimator for g(·) is proved to be weakly consistent with an optimal convergent rate.     

General admissibility for linear estimators in a general multivariate linear model under balanced loss function

A generalization of Zellner’s balanced loss function is proposed. General admissibility in a general multivariate linear model is investigated under the generalized balanced loss function. And the sufficient and necessary conditions for linear estimators to be generally admissible in classes of homogeneous and nonhomogeneous linear estimators are given, respectively.     

部分线性变系数模型Backfitting估计的渐近性质

作为部分线性模型与变系数模型的推广,部分线性变系数模型是一类应用广泛的数据分析模型.利用Backfitting方法拟合这类特殊的可加模型,可得到模型中常值系数估计量的精确解析表达式,该估计量被证明是n~(1/2)相合的.最后通过数值模拟考察了所提估计方法的有效性.     

Empirical likelihood-based inference in a partially linear model for longitudinal data

A partially linear model with longitudinal data is considered, empirical likelihood to infer- ence for the regression coefficients and the baseline function is investigated, the empirical log-likelihood ratios is proven to be asymptotically chi-squared, and the corresponding confidence regions for the pa- rameters of interest are then constructed. Also by the empirical likelihood ratio functions, we can obtain the maximum empirical likelihood estimates of the regression coefficients and the baseline function, and prove the asymptotic normality. The numerical results are conducted to compare the performance of the empirical likelihood and the normal approximation-based method, and a real example is analysed.     

The convergence rates of empirical Bayes estimation in a multiple linear regression model

Empirical Bayes (EB) estimation of the parameter vector =(,²) in a multiple linear regression modelY=X+ is considered, where is the vector of regression coefficient, N(0,² I) and ² is unknown. In this paper, we have constructed the EB estimators of by using the kernel estimation of multivariate density function and its partial derivatives. Under suitable conditions it is shown that the convergence rates of the EB estimators areO(n ^{-(k-1)(k-2)/k(2k+p+1)}), where the natural numberk3, 1/3<<1, andp is the dimension of vector .The project is supported by the National Natural Science Foundation of China.     

On the calculation of cumulants of estimators arising from a linear time series regression model

Bounds for higher-order cumulants of statistics arising from a linear time series regression model are investigated. A result given in Brillinger is proved and extended. The bounds permit derivation of asymptotic moments and asymptotic normality for estimators of parameters in the model. Two examples are given as illustrations.     

函数系数线性自回归模型的样条估计

基于多项式样条全局光滑方法,建立函数系数线性自回归模型中系数函数的样条估计.在适当条件下,证明了系数函数多项式样条估计的相合性,并给出了它们的收敛速度.模拟例子验证了理论结果的正确性.     

Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.     

Model checking for a general linear model with nonignorable missing covariates

In this paper, we investigate the model checking problem for a general linear model with nonignorable missing covariates. We show that, without any parametric model assumption for the response probability, the least squares method yields consistent estimators for the linear model even if only the complete data are applied. This makes it feasible to propose two testing procedures for the corresponding model checking problem: a score type lack-of-fit test and a test based on the empirical process. The asymptotic properties of the test statistics are investigated. Both tests are shown to have asymptotic power 1 for local alternatives converging to the null at the rate n-r, 0 ≤ r < 1/2 . Simulation results show that both tests perform satisfactorily.     

Asymptotic behaviour of Karmarkar's method for linear programming

The asymptotic behaviour of Karmarkar's method is studied and an estimate of the rate of the objective function value decrease is given. Two possible sources of numerical instability are discussed and a stabilizing procedure is proposed.Research supported in part by Republicka zajednica za nauku SR Srbije.     

New multiplier method for solving linear complementarity problems

A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. By introducing a Lagrangian of LCP(q, M), a new smooth merit function ϑ(x, λ) for LCP(q, M) is constructed. Based on it, a simple damped Newton-type algorithm with multiplier self-adjusting step is presented. When M is a P-matrix, the sequence {ϑ(x ^k, λ ^k)} (where {(x ^k, λ ^k)} is generated by the algorithm) is globally linearly convergent to zero and convergent in a finite number of iterations if the solution is degenerate. Numerical results suggest that the method is highly efficient and promising. Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(2): 162–171     

A class of estimators of the mean survival time from interval censored data with application to linear regression

A class of estimators of the mean survival time with interval censored data are studied by unbiased transformation method. The estimators are constructed based on the observations to ensure unbiasedness in the sense that the estimators in a certain class have the same expectation as the mean survival time. The estimators have good properties such as strong consistency （with the rate of O（n^-1/1 （log log n）^1/2）） and asymptotic normality. The application to linear regression is considered and the simulation reports are given.     

线性混合模型中方差分量的ANOVA估计的改进

讨论了在含三个方差分量的线性混合模型中,在均方误差意义下,方差分量的方差分析估计的改进,并把这一结果推广到一般的线性混合模型上,得到一个改进方差分析估计的简单方法.