Weighted estimating equation: modified GEE in longitudinal data analysis

The method of generalized estimating equations （GEE） introduced by K. Y. Liang and S. L. Zeger has been widely used to analyze longitudinal data. Recently, this method has been criticized for a failure to protect against misspecification of working correlation models, which in some cases leads to loss of efficiency or infeasibility of solutions. In this paper, we present a new method named as ＇weighted estimating equations （WEE）＇ for estimating the correlation parameters. The new estimates of correlation parameters are obtained as the solutions of these weighted estimating equations. For some commonly assumed correlation structures, we show that there exists a unique feasible solution to these weighted estimating equations regardless the correlation structure is correctly specified or not. The new feasible estimates of correlation parameters are consistent when the working correlation structure is correctly specified. Simulation results suggest that the new method works well in finite samples.     

Empirical likelihood inference for estimating equation with missing data

In this article, empirical likelihood inference for estimating equation with missing data is considered. Based on the weighted-corrected estimating function, an empirical log-likelihood ratio is proved to be a standard chi-square distribution asymptotically under some suitable conditions. This result is different from those derived before. So it is convenient to construct confidence regions for the parameters of interest. We also prove that our proposed maximum empirical likelihood estimator θ is asymptotically normal and attains the semiparametric efficiency bound of missing data. Some simulations indicate that the proposed method performs the best.     

Asymptotically self-similar solutions of the damped wave equation

We consider the damped hyperbolic equation

(1)     

Optimal and strongly optimal solutions for linear programming models with variable parameters

This paper deals with linear programming (LP) models with variable parameters and introduces two concepts for this class of problems: optimal solution and strongly optimal solution. Also, it seeks necessary and sufficient conditions for a feasible solution to be optimal or strongly optimal.     

Asymptotically optimal pointwise and minimax quickest change-point detection for dependent data

We consider the quickest change-point detection problem in pointwise and minimax settings for general dependent data models. Two new classes of sequential detection procedures associated with the maximal “local” probability of a false alarm within a period of some fixed length are introduced. For these classes of detection procedures, we consider two popular risks: the expected positive part of the delay to detection and the conditional delay to detection. Under very general conditions for the observations, we show that the popular Shiryaev–Roberts procedure is asymptotically optimal, as the local probability of false alarm goes to zero, with respect to both these risks pointwise (uniformly for every possible point of change) and in the minimax sense (with respect to maximal over point of change expected detection delays). The conditions are formulated in terms of the rate of convergence in the strong law of large numbers for the log-likelihood ratios between the “change” and “no-change” hypotheses, specifically as a uniform complete convergence of the normalized log-likelihood ratio to a positive and finite number. We also develop tools and a set of sufficient conditions for verification of the uniform complete convergence for a large class of Markov processes. These tools are based on concentration inequalities for functions of Markov processes and the Meyn–Tweedie geometric ergodic theory. Finally, we check these sufficient conditions for a number of challenging examples (time series) frequently arising in applications, such as autoregression, autoregressive GARCH, etc.     

Asymptotically self-similar global solutions of a general semilinear heat equation

We consider the general nonlinear heat equation on where and g satisfies certain growth conditions. We prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation. We also prove that some of those global solutions are asymptotic for large time to self-similar solutions of the single power nonlinear heat equation, i.e. with Received: 23 July 1999 / Accepted: 14 December 2000 / Published online: 23 July 2001     

Analytic solutions to a strongly nonlinear Vlasov equation

We prove the existence for short times of analytic solutions to a Vlasov type equation. The corresponding model is one-dimensional but uses a quite singular force term which involves a full derivative in x of the macroscopic density, making the existence of solutions a difficult question.     

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.     

Copula and composite quantile regression-based estimating equations for longitudinal data

Annals of the Institute of Statistical Mathematics - Composite quantile regression (CQR) is a powerful complement to the usual mean regression and becomes increasingly popular due to its robustness...     

New estimating equation approaches with application in lifetime data analysis

Estimating equation approaches have been widely used in statistics inference. Important examples of estimating equations are the likelihood equations. Since its introduction by Sir R. A. Fisher almost a century ago, maximum likelihood estimation (MLE) is still the most popular estimation method used for fitting probability distribution to data, including fitting lifetime distributions with censored data. However, MLE may produce substantial bias and even fail to obtain valid confidence intervals when data size is not large enough or there is censoring data. In this paper, based on nonlinear combinations of order statistics, we propose new estimation equation approaches for a class of probability distributions, which are particularly effective for skewed distributions with small sample sizes and censored data. The proposed approaches may possess a number of attractive properties such as consistency, sufficiency and uniqueness. Asymptotic normality of these new estimators is derived. The construction of new estimation equations and their numerical performance under different censored schemes are detailed via Weibull distribution and generalized exponential distribution.     

On regular solutions of a strongly degenerate elliptic equation

Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 2, pp. 1–20, April–June, 1995.     

Asymptotically optimal parallel resource assignment with interference

Motivated by scheduling in cellular wireless networks and resource allocation in computer systems, we study a service facility with two classes of users having heterogeneous service requirement distributions. The aggregate service capacity is assumed to be largest when both classes are served in parallel, but giving preferential treatment to one of the classes may be advantageous when aiming at minimization of the number of users, or when classes have different economic values, for example.     

Asymptotically optimal estimators for moments of change

We consider the problem of finding asymptotically optimal estimators for many moments of change in the case of incomplete information on distributions. We prove that if the maximum-likelihood estimator is asymptotically optimal, then, under certain conditions, it preserves this property after the replacement of actual values by density estimators. We solve the problem for the case of one moment of change and generalize the results obtained to the case of several moments of change. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 406–416, March, 2006.     

The semilinear second-order hyperbolic equation with data strongly singular at one point

In this paper we study the initial value problem for the scalar semilinear strictly hyperbolic equation in multidimensional space with data strongly singular at one point. Under the assumption of the initial data being conormal with respect to one point and bounded or regular with a certain low degree, the existence of the solution to this problem is obtained; meanwhile, it is proved that the singularity of the solution will spread on the forward characteristic cone of the hyperbolic operator issuing from this point, and the solution is bounded and conormal with respect to this cone.     

Regularization of a two-dimensional strongly damped wave equation with statistical discrete data

In this paper, we consider an inverse problem for a strongly damped wave equation in two dimensional with statistical discrete data. Firstly, we give a representation for the solution and then present a discretization form of the Fourier coefficients. Secondly, we show that the solution does not depend continuously on the data by stating a concrete example, which makes the solution be not stable and thus the present problem is ill-posed in the sense of Hadamard. Next, we use the trigonometric least squares method associated with the Fourier truncation method to regularize the instable solution of the problem. Finally, the convergence rate of the error between the regularized solution and the sought solution is estimated and also investigated numerically.     

Functional limit theorem for solutions to burgers equation with random initial data

A functional limit theorem is proved for solutions to Burgers’ equation corresponding to the sequence of initial potentials determined by specified shot noise random fields.