Fuzzy clusterwise quasi-likelihood generalized linear models

The quasi-likelihood method has emerged as a useful approach to the parameter estimation of generalized linear models (GLM) in circumstances where there is insufficient distributional information to construct a likelihood function. Despite its flexibility, the quasi-likelihood approach to GLM is currently designed for an aggregate-sample analysis based on the assumption that the entire sample of observations is taken from a single homogenous population. Thus, this approach may not be suitable when heterogeneous subgroups exist in the population, which involve qualitatively distinct effects of covariates on the response variable. In this paper, the quasi-likelihood GLM approach is generalized to a fuzzy clustering framework which explicitly accounts for such cluster-level heterogeneity. A simple iterative estimation algorithm is presented to optimize the regularized fuzzy clustering criterion of the proposed method. The performance of the proposed method in recovering parameters is investigated based on a Monte Carlo analysis involving synthetic data. Finally, the empirical usefulness of the proposed method is illustrated through an application to actual data on the coupon usage behaviour of a sample of consumers.     

Adaptive quasi-likelihood estimate in generalized linear models

This paper gives a thorough theoretical treatment on the adaptive quasi-likelihood estimate of the parameters in the generalized linear models. The unknown covariance matrix of the response variable is estimated by the sample. It is shown that the adaptive estimator defined in this paper is asymptotically most efficient in the sense that it is asymptotic normal, and the covariance matrix of the limit distribution coincides with the one for the quasi-likelihood estimator for the case that the covariance matrix of the response variable is completely known.     

A robust and efficient estimation method for single-index varying-coefficient models

A new estimation procedure based on modal regression is proposed for single-index varying-coefficient models. The proposed method achieves better robustness and efficiency than that of Xue and Pang (2013). We establish the asymptotic normalities of proposed estimators and evaluate the performance of the proposed method by a numerical simulation.     

Nonparametric iterative estimation of multivariate binary density

In this paper, an iterative estimate of the multivariate density is proposed when the variables are binary in nature. Some properties of this estimate are also discussed. Finally, applications of this estimate are discussed in the areas of pattern recognition and reliability.     

Strong consistency of maximum quasi-likelihood estimates in generalized linear models

In a generalized linear model with q×1 responses, bounded and fixed p×q regressors zi and general link function, under the most general assumption on the minimum eigenvalue of ∑in=1 ZiZi', the moment condition on responses as weak as possible and other mild regular conditions, we prove that with probability one, the quasi-likelihood equation has a solution βn for all large sample size n, which converges to the true regression parameter β0. This result is an essential improvement over the relevant results in literature.     

The quasi-likelihood estimation in regression

I propose a simply method to estimate the regression parameters in quasi-likelihood model My main approach utilizes the dimension reduction technique to first reduce the dimension of the regressor X to one dimension before solving the quasi-likelihood equations. In addition, the real advantage of using dimension reduction technique is that it provides a good initial estimate for one-step estimator of the regression parameters. Under certain design conditions, the estimators are asymptotically multivariate normal and consistent. Moreover, a Monte Carlo simulation is used to study the practical performance of the procedures, and I also assess the cost of CPU time for computing the estimates.This research partially supported by the National Science Council, R.O.C. (Plan No. NSC 82-0208-M-032-023-T).     

自适应设计广义线性模型极大拟似然估计的渐近性

在对Fisher信息矩阵的最小特征根最一般的假定,响应变量的矩条件尽可能弱和其它正则条件下,证明了自适应设计广义线性模型中极大拟似然估计的强相合性与渐近正态性,同时给出了强收敛速度.     

Asymptotic normality and strong consistency of maximum quasi-likelihood estimates in generalized linear models

In a generalized linear model with q x 1 responses, the bounded and fixed (or adaptive) p × q regressors Zi and the general link function, under the most general assumption on the minimum eigenvalue of ZiZ'i,the moment condition on responses as weak as possible and the other mild regular conditions, we prove that the maximum quasi-likelihood estimates for the regression parameter vector are asymptotically normal and strongly consistent.     

Nonparametric regression estimation in models with weak error's structure

In this paper we propose nonparametric estimates of the regression function and its derivative when it is only assumed a weak error's structure. We study their local and global asymptotic behaviour when we observe dependent trajectories.     

Nonparametric estimation of competing risks models with covariates

In competing risks model, several failure times arise potentially. The smallest failure time and its index only are observed. Without specific assumptions, the joint or even the marginal distribution functions of the underlying failure times are not identifiable (A. Tsiatis, Proc. Natl. Acad. Sci. USA 72 (1975) 20). Nonetheless, if each individual is characterized by a “sufficiently informative” set of covariates, these distributions are identifiable under some conditions of regularity (J.J. Heckman and B. Honoré, Biometrika 76 (1989) 325). In this paper, nonparametric kernel estimators of the joint distribution function of failure times conditional on the covariates are proposed. Their weak and strong consistency are discussed.     

Rate of strong consistency of the maximum quasi-likelihood estimator in quasi-likelihood nonlinear models

Quasi-likelihood nonlinear models （QLNM） include generalized linear models as a special case. Under some regularity conditions, the rate of the strong consistency of the maximum quasi-likelihood estimation （MQLE） is obtained in QLNM. In an important case, this rate is O（n-^1/2（loglogn）^1/2）, which is just the rate of LIL of partial sums for i.i.d variables, and thus cannot be improved anymore.     

Nonparametric estimation of the location and scale parameters based on density estimation

Summary By representing the location and scale parameters of an absolutely continuous distribution as functionals of the usually unknown probability density function, it is possible to provide estimates of these parameters in terms of estimates of the unknown functionals. Using the properties of well-known methods of density estimates, it is shown that the proposed estimates possess nice large sample properties and it is indicated that they are also robust against dependence in the sample. The estimates perform well against other estimates of location and scale parameters.     

Parameter estimation in linear static systems based on weighted least-absolute value estimation

In this paper, we present a new method for estimating a parameter vector of which measurements are carried out; however, these measurements are subjected to noise. First, we briefly consider least-square estimation of such vector to obtain some well-known results. Then, we proceed to formulate the problem in the least-absolute value (LAV) sense and show that we can obtain a set of overdetermined equations for the components of the unknown vector. These equations are solved using the least-square approach to ascertain which points give the least residuals. Having gained that information, we set to zero a number of residuals equal to the rank of the matrixH. Let this rank bek; then, the number of points which satisfy the LAV solution identically isk; this is a requirement that the LAV solution must satisfy (Refs. 1, 2). Several examples are presented in the paper.This work was supported by the Natural Science and Engineering Research Council of Canada, Grant A4146.     

An iterative method for generalized nonlinear complementarity problems

An iterative method for solving generalized nonlinear complementarity problems (Ref. 1) involving stronglyK-copositive operators is introduced. Conditions are presented which guarantee the convergence of the method; in addition, the sequence of iterates is used to prove the existence of a solution to the problem under conditions not included in the previous study. Separate consideration is given to the generalized linear complementarity problem.This research was partially supported by National Science Foundation, Grant No. GP-16293. This paper constitutes part of the junior author's doctoral thesis written at Rensselaer Polytechnic Institute. Research support was provided by an NDEA Fellowship and an RPI Fellowship.     

Consistency and asymptotic normality of the maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with random regressors

This paper proposes some regularity conditions, which result in the existence, strong consistency and asymptotic normality of maximum quasi-likelihood estimator （MQLE） in quasi-likelihood nonlinear models （QLNM） with random regressors. The asymptotic results of generalized linear models （GLM） with random regressors are generalized to QLNM with random regressors.     

Robust construction of regression models based on the generalized least absolute deviations method

We suggest a new method of robust construction of linear regression models, namely, the generalized least absolute deviations method. We give a theoretical justification of the method and consider its experimental approbation. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 236–250.     

LSQR iterative method for generalized coupled Sylvester matrix equations

An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and (R, S)-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method.     

Nonparametric estimation based on censored observations of a Markov renewal process

Summary Uniform consistency and weak convergence is proved of estimators of the transition probabilities of an arbitrary finite state space Markov renewal process, based on n independent and identically distributed right censored realizations of the process. The approach uses the theory of stochastic integrals and counting processes. It is shown how the results may be extended to the non-identically distributed case and to general censorship under suitable conditions.