An embedded estimating equation for the additive risk model with biased-sampling data

This paper presents a novel class of semiparametric estimating functions for the additive model with right-censored data that are obtained from general biased-sampling. The new estimator can be obtained using a weighted estimating equation for the covariate coeffcients, by embedding the biased-sampling data into left-truncated and right-censored data. The asymptotic properties (consistency and asymptotic normality) of the proposed estimator are derived via the modern empirical processes theory. Based on the cumulative residual processes, we also propose graphical and numerical methods to assess the adequacy of the additive risk model. The good finite-sample performance of the proposed estimator is demonstrated by simulation studies and two applications of real datasets.     

Asymptotic behavior of the threshold minimizing the average probability of error in calculation of wavelet coefficients

The problem of estimating the signal function from noisy observations by thresholding the coefficients of its wavelet decomposition is considered. The asymptotic orders of the threshold and risk are calculated by minimizing the average probability of error in calculating the wavelet coefficients.     

Asymptotic lower bounds for the risk of estimators of the value of a spectral density function

Summary Discrete parameter stationary processes with joint Gaussian distributions are considered. Loss is measured by squared error. It is shown that when estimating the value of g(x ₀) of a spectral density function g the risk has an asymptotic lower bound similar to those previously obtained by Farrell [6] for the problem of density function estimation. A similar result is obtained for estimates of an entire function when risk is measured by integrated square error.     

信用传染违约Aalen加性风险模型

本文考虑了基于加性风险模型的信用风险违约预报模型,不但考虑了宏观因素和公司个体因素,并且通过引入行业因素来刻画公司间可能存在的不同于宏观因素的信用传染效应,由此克服了以往模型对违约相关性的低估.本文在参数加性风险模型下给出极大似然估计及渐近性,提出两种估计方法并比较二者表现,得到最优权估计更加有效.同时本文还考虑了半参数的风险模型,并基于鞅的估计方程得到其估计及渐近性,均得到不错的结果.     

Nonparametric asymptotically efficient estimation of a signal in the nonlinear case

The statistical problem of estimating an unknown function α(t) from the observation process y(t)=AS(t, α(t))+n(t) is investigated, where n(t) is white noise and no prior information about α(t) is assumed. A Cramér-Rao bound for estimating α(t) is obtained and two constructive procedures are presented to produce nonanticipative nonparametric estimators on the basis of the likelihood equation in a function space. Consistency, asymptotic efficiency, asymptotic minimaxity and asymptotic sufficiency of the constructed estimator are proved. The extensive applicability of the nonparametric estimate is shown by applying the theory to the reception of a phase-modulated signal.     

Adaptive varying-coefficient linear quantile model: a profiled estimating equations approach

We consider an estimating equations approach to parameter estimation in adaptive varying-coefficient linear quantile model. We propose estimating equations for the index vector of the model in which the unknown nonparametric functions are estimated by minimizing the check loss function, resulting in a profiled approach. The estimating equations have a bias-corrected form that makes undersmoothing of the nonparametric part unnecessary. The estimating equations approach makes it possible to obtain the estimates using a simple fixed-point algorithm. We establish asymptotic properties of the estimator using empirical process theory, with additional complication due to the nuisance nonparametric part. The finite sample performance of the new model is illustrated using simulation studies and a forest fire dataset.     

Estimating function with asymptotic bias and its estimator

We introduce the estimating function with asymptotic bias and investigate the asymptotic behavior of the estimator based on it by using their relationship. The estimator based on the estimating function with asymptotic bias has the asymptotic normality with asymptotic bias. We show that this theory has several interesting applications in practical statistics.     

Estimate for Euclidean parameters of a mixture of two symmetric distributions

A sample from a mixture of two symmetric distributions is observed. The considered distributions differ only by a shift. Estimates are constructed by the method of estimating equations for parameters of mean locations and concentrations (mixing probabilities) of both components. We obtain conditions for the asymptotic normality of these estimates. The greatest lower bounds for the coefficients of dispersion of the estimates are determined.     

Improvement of estimators in a linear regression problem with random errors in coefficients

Under consideration is the problem of estimating the linear regression parameter in the case when the variances of observations depend on the unknown parameter of the model, while the coefficients (independent variables) are measured with random errors. We propose a new two-step procedure for constructing estimators which guarantees their consistency, find general necessary and sufficient conditions for the asymptotic normality of these estimators, and discuss the case in which these estimators have the minimal asymptotic variance.     

A modification of Miller's recurrence algorithm

Miller's recurrence algorithm for tabulating the subdominant solution of a second-order difference equation is modified so as to take the asymptotic behaviour of the solution into account. The asymptotic solutions of various types of equations are listed, and a method is given for estimating the error in the tabulated solution.     

带相依终止事件的复发事件数据的可加可乘比率模型

本文对带相依终止事件的复发事件数据提出了一个联合建模分析方法,用一个带脆弱变量的可加可乘比率模型来刻画复发事件过程,还用带脆弱变量的Cox风险率模型来刻画终止事件过程,而且这两个过程的相依性由脆弱变量来刻画.我们利用估计方程的方法,对模型参数进行了估计,给出了所得估计的渐近性质.同时,通过数值模拟分析验证了估计的渐近性质.最后,利用该方法分析了弗吉尼亚大学慢性心脏病病人医疗诊费数据.     

Estimation of a time series model from unequally spaced data

A process generated by a stochastic differential equation driven by pure noise is sampled at irregular intervals. A model for the sampled sequence is deduced. We describe a maximum likelihood procedure for estimating the parameters and establish the strong consistency and asymptotic normality of the estimates. The use of the model in prediction is considered. Simplifications in the case of periodic sampling are explored.     

Variable selection in the partially linear errors-in-variables models for longitudinal data

This paper proposes a new approach for variable selection in partially linear errors-in-variables (EV) models for longitudinal data by penalizing appropriate estimating functions. We apply the SCAD penalty to simultaneously select significant variables and estimate unknown parameters. The rate of convergence and the asymptotic normality of the resulting estimators are established. Furthermore, with proper choice of regularization parameters, we show that the proposed estimators perform as well as the oracle procedure. A new algorithm is proposed for solving penalized estimating equation. The asymptotic results are augmented by a simulation study.     

Matrix shrinkage of high-dimensional expectation vectors

The shrinkage effect is studied in estimating the expectation vector by weighting of mean vector components in the system of coordinates in which sample covariance matrix is diagonal. The Kolmogorov asymptotic approach is applied, when sample size increases together with the dimension, so that their ratio tends to a constant. Under some weak assumptions on the dependence of variables, the limit expression for the principal part of the quadratic risk function is found in dependence of weighting function. It is proved that the limit risk function does not depend on distributions. The extremum problem is solved, and an approximately unimprovable distribution-free estimator of the expectation vector is proposed.     

Parameter estimation for a stratified finite population

An asymptotic approach is proposed to estimating the class sizes in a finite population.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 11–14, 1986.     

Asymptotically optimal estimation in a linear regression problem with random errors in coefficients

We consider the problem of estimating an unknown one-dimensional parameter in the linear regression problem in the case when the independent variables (called coefficients in the article) are measured with errors, and the variances of the principal observations can depend on the main parameter. We study the behavior of two-step estimators, previously introduced by the authors, which are asymptotically optimal in the case when the independent variables are measured without errors. Under sufficiently general assumptions we find necessary and sufficient conditions for the asymptotic normality and asymptotic optimality of these estimators in the new setup.     

Godambe estimating functions and asymptotic optimal inference

Godambe (1985) introduced a class of optimum estimating functions which can be regarded as a generalization of quasilikelihood score functions. The “optimality” established by Godambe (1985) within a certain class is for estimating functions and it is based on finite samples. The question that arises naturally is what (if any) asymptotic optimality properties do the estimators and tests based on optimum estimating functions possess. In this paper, we establish, via presenting a convolution theorem, asymptotic optimality of estimators and tests obtained from Godambe optimum estimating functions. It is noted that we do not require the knowledge of the likelihood function.     

Analyzing the general biased data by additive risk model

This paper proposes a unified semiparametric method for the additive risk model under general biased sampling. By using the estimating equation approach, we propose both estimators of the regression parameters and nonparametric function. An advantage is that our approach is still suitable for the lengthbiased data even without the information of the truncation variable. Meanwhile, large sample properties of the proposed estimators are established, including consistency and asymptotic normality. In addition, the finite sample behavior of the proposed methods and the analysis of three groups of real data are given.     

Convergence of osculatory quadrature formulae

The regions of analyticity of functions to be integrated using equally spaced osculatory quadrature formulae are obtained. As a by-product it is noted that the asymptotic forms used are applicable to estimating or placing bounds on errors.     

Multivariate sequential point estimation

For a multivariate normal distribution with unknown mean vector and unknown dispersion matrix, a sequential procedure for estimating the unknown mean vector is suggested. The procedure is shown to be asymptotically “risk efficient” in the sense of Starr (Ann. Math. Statist. (1966), 1173–1185), and the asymptotic order of the “regret” (see Starr and Woodroofe, Proc. Nat. Acad. Sci. 63 (1969), 285–288) is given. Moderate sample behaviour of the procedure using Monte-Carlo techniques is also studied. Finally, the asymptotic normality of the stopping time is proved.